Ideal (order theory) 2From Wikipedia, the free encyclopedia

Contents

1 Annihilator (ring theory) 11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Chain conditions on annihilator ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Category-theoretic description for commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Relations to other properties of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Ascending chain condition on principal ideals 42.1 Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Associated prime 63.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Augmentation ideal 94.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Fractional ideal 115.1 Definition and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 Divisorial ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6 Going up and going down 136.1 Going up and going down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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6.1.1 Lying over and incomparability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.1.2 Going-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.1.3 Going down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.2 Going-up and going-down theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Ideal (order theory) 167.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.2 Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.3 Maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.6 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 Ideal (ring theory) 198.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.6 Ideal generated by a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.7 Types of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.8 Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.9 Ideal operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.10 Ideals and congruence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9 Ideal class group 259.1 History and origin of the ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.4 Relation with the group of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.5 Examples of ideal class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9.5.1 Class numbers of quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.6 Connections to class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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9.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10 Ideal norm 2910.1 Relative norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.2 Absolute norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

11 Ideal quotient 3111.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.2 Calculating the quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.3 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

12 Ideal theory 3312.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.2 In political philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

13 Jacobian ideal 3413.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

14 Jacobson radical 3514.1 Intuitive discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.2 Equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

15 Krull’s principal ideal theorem 3915.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

16 Krull’s theorem 4016.1 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4016.2 Krull’s Hauptidealsatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4016.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4016.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

17 Maximal ideal 4217.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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17.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

18 Minimal ideal 4518.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4618.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4618.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

19 Minimal prime ideal 4719.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

20 Nil ideal 4920.1 Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4920.2 Noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4920.3 Relation to nilpotent ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4920.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5020.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5020.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

21 Nilpotent ideal 5121.1 Relation to nil ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

22 Nilradical of a ring 5322.1 Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5322.2 Noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5322.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

23 Primary ideal 5523.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5523.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5623.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5623.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

24 Prime ideal 5724.1 Prime ideals for commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

24.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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24.1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5824.1.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

24.2 Prime ideals for noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

24.3 Important facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.4 Connection to maximality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6024.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6024.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

25 Primitive ideal 6225.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

26 Principal ideal 6326.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.2 Examples of non-principal ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.3 Related definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6426.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

27 Principal ideal theorem 6527.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

28 Radical of a ring 6728.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6728.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

28.2.1 The Jacobson radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6728.2.2 The Baer radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.2.3 The upper nil radical or Köthe radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.2.4 Singular radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.2.5 The Levitzki radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.2.6 The Brown–McCoy radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.2.7 The von Neumann regular radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6928.2.8 The Artinian radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

28.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6928.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

29 Radical of an ideal 7029.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7029.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7029.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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29.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7129.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7229.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7229.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

30 Regular ideal 7330.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

30.1.1 Modular ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7330.1.2 Regular element ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7330.1.3 Von Neumann regular ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.1.4 Quotient von Neumann regular ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

30.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

31 Semiprime ring 7631.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7631.2 General properties of semiprime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.3 Semiprime Goldie rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

32 Spectrum of a ring 7932.1 Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7932.2 Sheaves and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7932.3 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.4 Motivation from algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.5 Global Spec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.6 Representation theory perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.7 Functional analysis perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8232.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8232.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8232.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

33 Tight closure 8333.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8333.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 84

33.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8433.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8633.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Chapter 1

Annihilator (ring theory)

Inmathematics, specificallymodule theory, the annihilator of a set is a concept generalizing torsion and orthogonality.

1.1 Definitions

Let R be a ring, and letM be a left R-module. Choose a nonempty subset S ofM. The annihilator, denoted AnnR(S),of S is the set of all elements r in R such that for each s in S, rs = 0.[1] In set notation,

AnnR(S) = {r ∈ R | ∀s ∈ S, rs = 0}

It is the set of all elements of R that “annihilate” S (the elements for which S is torsion). Subsets of right modulesmay be used as well, after the modification of "sr = 0” in the definition.The annihilator of a single element x is usually written AnnR(x) instead of AnnR({x}). If the ring R can be understoodfrom the context, the subscript R can be omitted.Since R is a module over itself, S may be taken to be a subset of R itself, and since R is both a right and a left Rmodule, the notation must be modified slightly to indicate the left or right side. Usually ℓ.AnnR(S) and r.AnnR(S)or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary.If M is an R-module and AnnR(M) = 0, then M is called a faithful module.

1.2 Properties

If S is a subset of a left R module M, then Ann(S) is a left ideal of R. The proof is straightforward: If a and b bothannihilate S, then for each s in S, (a + b)s = as + bs = 0, and for any r in R, (ra)s = r(as) = r0 = 0. (A similar prooffollows for subsets of right modules to show that the annihilator is a right ideal.)If S is a submodule of M, then AnnR(S) is even a two-sided ideal: (ac)s = a(cs) = 0, since cs is another element ofS.[2]

If S is a subset of M and N is the submodule of M generated by S, then in general AnnR(N) is a subset of AnnR(S),but they are not necessarily equal. If R is commutative, then it is easy to check that equality holds.M may be also viewed as a R/AnnR(M)-module using the action rm := rm . Incidentally, it is not always possibleto make an Rmodule into an R/I module this way, but if the ideal I is a subset of the annihilator ofM, then this actionis well defined. Considered as an R/AnnR(M)-module, M is automatically a faithful module.

1.3 Chain conditions on annihilator ideals

The lattice of ideals of the form ℓ.AnnR(S) where S is a subset ofR comprise a complete lattice when partially orderedby inclusion. It is interesting to study rings for which this lattice (or its right counterpart) satisfy the ascending chain

1

2 CHAPTER 1. ANNIHILATOR (RING THEORY)

condition or descending chain condition.Denote the lattice of left annihilator ideals of R as LA and the lattice of right annihilator ideals of R as RA . Itis known that LA satisfies the A.C.C. if and only if RA satisfies the D.C.C., and symmetrically RA satisfies theA.C.C. if and only if LA satisfies the D.C.C. If either lattice has either of these chain conditions, then R has noinfinite orthogonal sets of idempotents. (Anderson 1992, p.322) (Lam 1999)If R is a ring for which LA satisfies the A.C.C. and RR has finite uniform dimension, then R is called a left Goldiering. (Lam 1999)

1.4 Category-theoretic description for commutative rings

When R is commutative and M is an R-module, we may describe AnnR(M) as the kernel of the action map R →EndR(M) determined by the adjunct map of the identity M → M along the Hom-tensor adjunction.More generally, given a bilinear map of modules F : M ×N → P , the annihilator of a subset S ⊂ M is the set ofall elements in N that annihilate S :

Ann(S) := {n ∈ N | ∀s ∈ S, F (s, n) = 0}.

Conversely, given T ⊂ N , one can define an annihilator as a subset ofM .The annihilator gives a Galois connection between subsets of M and N , and the associated closure operator isstronger than the span. In particular:

• annihilators are submodules

• Span(S) ≤ Ann(Ann(S))

• Ann(Ann(Ann(S))) = Ann(S)

An important special case is in the presence of a nondegenerate form on a vector space, particularly an inner product:then the annihilator associated to the map V × V → K is called the orthogonal complement.

1.5 Relations to other properties of rings• Annihilators are used to define left Rickart rings and Baer rings.

• The set of (left) zero divisors DS of S can be written as

DS =∪

x∈S, x =0

AnnR (x).

(Here we allow zero to be a zero divisor.)

In particularDR is the set of (left) zero divisors ofR taking S =R andR acting on itself as a leftR-module.

• When R is commutative and Noetherian, the set DR is precisely equal to the union of the associated primeideals of R.

1.6 See also• socle

1.7. NOTES 3

1.7 Notes[1] Pierce (1982), p. 23.

[2] Pierce (1982), p. 23, Lemma b, item (i).

1.8 References• Weisstein, Eric W., “Annihilator”, MathWorld.

• Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics13 (2 ed.), New York: Springer-Verlag, pp. x+376, ISBN 0-387-97845-3, MR 1245487

• Israel Nathan Herstein (1968) Noncommutative Rings, Carus Mathematical Monographs #15, MathematicalAssociation of America, page 3.

• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, NewYork: Springer-Verlag, pp. 228–232, ISBN 978-0-387-98428-5, MR 1653294

• Richard S. Pierce. Associative algebras. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, ISBN978-0-387-90693-5

Chapter 2

Ascending chain condition on principalideals

In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, orprincipal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principalideals (abbreviated toACCP) is satisfied if there is no infinite strictly ascending chain of principal ideals of the giventype (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.The counterpart descending chain condition may also be applied to these posets, however there is currently no needfor the terminology “DCCP” since such rings are already called left or right perfect rings. (See Noncommutative ringsection below.)Noetherian rings (e.g. principal ideal domains) are typical examples, but some important non-Noetherian rings alsosatisfy (ACCP), notably unique factorization domains and left or right perfect rings.

2.1 Commutative rings

It is well known that a nonzero nonunit in a Noetherian integral domain factors into irreducibles. The proof of thisrelies on only (ACCP) not (ACC), so in any integral domain with (ACCP), an irreducible factorization exists. (Inother words, any integral domains with (ACCP) are atomic. But the converse is false, as shown in (Grams 1974).)Such a factorization may not be unique; the usual way to establish uniqueness of factorizations uses Euclid’s lemma,which requires factors to be prime rather than just irreducible. Indeed one has the following characterization: let Abe an integral domain. Then the following are equivalent.

1. A is a UFD.

2. A satisfies (ACCP) and every irreducible of A is prime.

3. A is a GCD domain satisfying (ACCP).

The so-called Nagata criterion holds for an integral domain A satisfying (ACCP): Let S be a multiplicatively closedsubset of A generated by prime elements. If the localization S−1A is a UFD, so is A. (Nagata 1975, Lemma 2.1)(Note that the converse of this is trivial.)An integral domain A satisfies (ACCP) if and only if the polynomial ring A[t] does.[1] The analogous fact is false ifA is not an integral domain. (Heinzer, Lantz 1994)An integral domain where every finitely generated ideal is principal (that is, a Bézout domain) satisfies (ACCP) ifand only if it is a principal ideal domain.[2]

The ringZ+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actuallya GCD domain) that does not satisfy (ACCP), for the chain of principal ideals

(X) ⊂ (X/2) ⊂ (X/4) ⊂ (X/8), ...

4

2.2. NONCOMMUTATIVE RINGS 5

is non-terminating.

2.2 Noncommutative rings

In the noncommutative case, it becomes necessary to distinguish the right ACCP from left ACCP. The former onlyrequires the poset of ideals of the form xR to satisfy the ascending chain condition, and the latter only examines theposet of ideals of the form Rx.A theorem of Hyman Bass in (Bass 1960) now known as “Bass’ Theorem P” showed that the descending chaincondition on principal left ideals of a ring R is equivalent to R being a right perfect ring. D. Jonah showed in (Jonah1970) that there is a side-switching connection between the ACCP and perfect rings. It was shown that if R is rightperfect (satisfies right DCCP), then R satisfies the left ACCP, and symmetrically, if R is left perfect (satisfies leftDCCP), then it satisfies the right ACCP. The converses are not true, and the above switches from “left” and “right”are not typos.Whether the ACCP holds on the right or left side of R, it implies that R has no infinite set of nonzero orthogonalidempotents, and that R is a Dedekind finite ring. (Lam 1999, p.230-231)

2.3 References[1] Gilmer, Robert (1986), “Property E in commutative monoid rings”, Group and semigroup rings (Johannesburg, 1985),

North-Holland Math. Stud. 126, Amsterdam: North-Holland, pp. 13–18, MR 860048.

[2] Proof: In a Bézout domain the ACCP is equivalent to the ACC on finitely generated ideals, but this is known to be equivalentto the ACC on all ideals. Thus the domain is Noetherian and Bézout, hence a principal ideal domain.

• Bass, Hyman (1960), “Finitistic dimension and a homological generalization of semi-primary rings”, Trans.Amer. Math. Soc. 95: 466–488, doi:10.1090/s0002-9947-1960-0157984-8, ISSN 0002-9947, MR 0157984

• Grams, Anne (1974), “Atomic rings and the ascending chain condition for principal ideals”, Proc. CambridgePhilos. Soc. 75: 321–329, doi:10.1017/s0305004100048532, MR 0340249

• Heinzer, William J.; Lantz, David C. (1994), “ACCP in polynomial rings: a counterexample”, Proc. Amer.Math. Soc. 121 (3): 975–977, doi:10.2307/2160301, ISSN 0002-9939, JSTOR 2160301, MR 1653294

• Jonah, David (1970), “Rings with theminimum condition for principal right ideals have themaximum conditionfor principal left ideals”, Math. Z. 113: 106–112, doi:10.1007/bf01141096, ISSN 0025-5874, MR 0260779

• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

• Nagata, Masayoshi (1975), “Some types of simple ring extensions”, Houston J. Math. 1 (1): 131–136, ISSN0362-1588, MR 0382248

Chapter 3

Associated prime

In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as anannihilator of a (prime) submodule of M. The set of associated primes is usually denoted by AssR(M) .In commutative algebra, associated primes are linked to the Lasker-Noether primary decomposition of ideals incommutative Noetherian rings. Specifically, if an ideal J is decomposed as a finite intersection of primary ideals,the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with AssR(R/J) .[1]Also linked with the concept of “associated primes” of the ideal are the notions of isolated primes and embeddedprimes.

3.1 Definitions

A nonzero R module N is called a prime module if the annihilator AnnR(N) = AnnR(N ′) for any nonzerosubmodule N' of N. For a prime module N, AnnR(N) is a prime ideal in R.[2]

An associated prime of an R module M is an ideal of the form AnnR(N) where N is a prime submodule of M. Incommutative algebra the usual definition is different, but equivalent:[3] if R is commutative, an associated prime Pof M is a prime ideal of the form AnnR(m) for a nonzero element m of M or equivalently R/P is isomorphic to asubmodule of M.In a commutative ring R, minimal elements in AssR(M) (with respect to the set-theoretic inclusion) are called iso-lated primes while the rest of the associated primes (i.e., those properly containing associated primes) are calledembedded prime.A module is called coprimary if xm = 0 for some nonzero m ∈ M implies xnM = 0 for some positive integer n. Anonzero finitely generated moduleM over a commutative Noetherian ring is coprimary if and only if it has exactly oneassociated prime. A submodule N of M is called P-primary ifM/N is coprimary with P. An ideal I is a P-primaryideal if and only if AssR(R/I) = {P} ; thus, the notion is a generalization of a primary ideal.

3.2 Properties

Most of these properties and assertions are given in (Lam 2001) starting on page 86.

• If M' ⊆M, then AssR(M ′) ⊆ AssR(M) . If in addition M' is an essential submodule of M, their associatedprimes coincide.

• It is possible, even for a commutative local ring, that the set of associated primes of a finitely generated moduleis empty. However, in any ring satisfying the ascending chain condition on ideals (for example, any right orleft Noetherian ring) every nonzero module has at least one associated prime.

• Any uniform module has either zero or one associated primes, making uniform modules an example of copri-mary modules.

6

3.3. EXAMPLES 7

• For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposableinjective modules onto the spectrum Spec(R) . If R is an Artinian ring, then this map becomes a bijection.

• Matlis’ Theorem: For a commutative Noetherian ring R, the map from the isomorphism classes of indecom-posable injective modules to the spectrum is a bijection. Moreover, a complete set of representatives for thoseclasses is given by E(R/p) where E(−) denotes the injective hull and p ranges over the prime ideals of R.

• For a Noetherian module M over any ring, there are only finitely many associated primes of M.

The following properties all refer to a commutative Noetherian ring R:

• Every ideal J (through primary decomposition) is expressible as a finite intersection of primary ideals. Theradical of each of these ideals is a prime ideal, and these primes are exactly the elements of AssR(R/J) . Inparticular, an ideal J is a primary ideal if and only if AssR(R/J) has exactly one element.

• Any prime ideal minimal with respect to containing an ideal J is in AssR(R/J) . These primes are preciselythe isolated primes.

• The set theoretic union of the associated primes of M is exactly the collection of zero-divisors on M, that is,elements r for which there exists nonzero m in M with mr =0.

• If M is a finitely generated module over R, then there is a finite ascending sequence of submodules

0 = M0 ⊂ M1 ⊂ · · · ⊂ Mn−1 ⊂ Mn = M

such that each quotient Mi/Mi−1 is isomorphic to R/Pi for some prime ideals Pi. Moreover everyassociated prime ofM occurs among the set of primes Pi. (In general not all the ideals Pi are associatedprimes of M.)

• Let S be a multiplicatively closed subset of R and f : Spec(S−1R) → Spec(R) the canonical map. Then, fora module M over R,

AssR(S−1M) = f(AssS−1R(S−1M)) = AssR(M) ∩ {P |P ∩ S = ∅} .[4]

• For a module M over R, Ass(M) ⊆ Supp(M) . Furthermore, the set of minimal elements of Supp(M)coincides with the set of minimal elements of Ass(M) . In particular, the equality holds if Ass(M) consistsof maximal ideals.

• A module M over R has finite length if and only if M is finitely generated and Ass(M) consists of maximalideals.[5]

3.3 Examples

• If R is the ring of integers, then non-trivial free abelian groups and non-trivial abelian groups of prime powerorder are coprimary.

• If R is the ring of integers andM a finite abelian group, then the associated primes ofM are exactly the primesdividing the order of M.

• The group of order 2 is a quotient of the integers Z (considered as a free module over itself), but its associatedprime ideal (2) is not an associated prime of Z.

8 CHAPTER 3. ASSOCIATED PRIME

3.4 References[1] Lam 1999, p. 117, Ex 40B.

[2] Lam 1999, p. 85.

[3] Lam 1999, p. 86.

[4] Matsumura 1970, 7.C Lemma

[5] Cohn, P. M. (2003), Basic Algebra, Springer, Exercise 10.9.7, p. 391, ISBN 9780857294289.

• Eisenbud, David (1995), Commutative algebra, Graduate Texts inMathematics 150, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960

• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

• Matsumura, Hideyuki (1970), Commutative algebra

Chapter 4

Augmentation ideal

In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If G is a group and R acommutative ring, there is a ring homomorphism ε , called the augmentation map, from the group ring

R[G]

to R, defined by taking a sum

∑rigi

to

∑ri.

Here ri is an element of R and gi an element of G. The sums are finite, by definition of the group ring. In less formalterms,

ε(g)

is defined as 1R whatever the element g in G, and ε is then extended to a homomorphism of R-modules in the obviousway. The augmentation ideal is the kernel of ε , and is therefore a two-sided ideal in R[G]. It is generated by thedifferences

g − g′

of group elements.Furthermore it is also generated by

g − 1, g ∈ G

which is a basis for the augmentation ideal as a free R module.For R andG as above, the group ring R[G] is an example of an augmented R-algebra. Such an algebra comes equippedwith a ring homomorphism to R. The kernel of this homomorphism is the augmentation ideal of the algebra.Another class of examples of augmentation ideal can be the kernel of the counit ε of any Hopf algebra.The augmentation ideal plays a basic role in group cohomology, amongst other applications.

9

10 CHAPTER 4. AUGMENTATION IDEAL

4.1 References• D. L. Johnson (1990). Presentations of groups. London Mathematical Society Student Texts 15. CambridgeUniversity Press. pp. 149–150. ISBN 0-521-37203-8.

• Dummit and Foote, Abstract Algebra

Chapter 5

Fractional ideal

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context ofintegral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of anintegral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ringideals are both under discussion, the latter are sometimes termed integral ideals for clarity.

5.1 Definition and basic results

Let R be an integral domain, and let K be its field of fractions. A fractional ideal of R is an R-submodule I of K suchthat there exists a non-zero r ∈ R such that rI ⊆ R. The element r can be thought of as clearing out the denominatorsin I. The principal fractional ideals are those R-submodules of K generated by a single nonzero element of K. Afractional ideal I is contained in R if, and only if, it is an ('integral') ideal of R.A fractional ideal I is called invertible if there is another fractional ideal J such that IJ = R (where IJ = { a1b1 + a2b2+ ... + anbn : ai ∈ I, bi ∈ J, n ∈ Z>₀ } is called the product of the two fractional ideals). In this case, the fractionalideal J is uniquely determined and equal to the generalized ideal quotient

(R : I) = {x ∈ K : xI ⊆ R}.

The set of invertible fractional ideals form an abelian group with respect to above product, where the identity is theunit ideal R itself. This group is called the group of fractional ideals of R. The principal fractional ideals form asubgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an R-module.Every finitely generated R-submodule of K is a fractional ideal and if R is noetherian these are all the fractional idealsof R.

5.2 Dedekind domains

In Dedekind domains, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. Infact, this property characterizes Dedekind domains: an integral domain is a Dedekind domain if, and only if, everynon-zero fractional ideal is invertible.The quotient group of fractional ideals by the subgroup of principal fractional ideals is an important invariant of aDedekind domain called the ideal class group.

5.3 Divisorial ideal

Let I denote the intersection of all principal fractional ideals containing a nonzero fractional ideal I. Equivalently,

11

12 CHAPTER 5. FRACTIONAL IDEAL

I = (R : (R : I)),

where as above

(R : I) = {x ∈ K : xI ⊆ R}.

If I = I then I is called divisorial.[1] In other words, a divisorial ideal is a nonzero intersection of some nonemptyset of fractional principal ideals. If I is divisorial and J is a nonzero fractional ideal, then (I : J) is divisorial.Let R be a local Krull domain (e.g., a Noetherian integrally closed local domain). Then R is a discrete valuation ringif and only if the maximal ideal of R is divisorial.[2]

An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.[3]

5.4 See also• divisorial sheaf

5.5 Notes[1] Bourbaki 1998, §VII.1

[2] Bourbaki Ch. VII, § 1, n. 7. Proposition 11.

[3] http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdffirstpage_1&handle=euclid.rmjm/1187453107

5.6 References• Chapter 9 of Atiyah, Michael Francis; Macdonald, I.G. (1994), Introduction to Commutative Algebra, WestviewPress, ISBN 978-0-201-40751-8

• Chapter VII.1 of Bourbaki, Nicolas (1998), Commutative algebra (2nd ed.), Springer Verlag, ISBN 3-540-64239-0

• Chapter 11 of Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Math-ematics 8 (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461

Chapter 6

Going up and going down

In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain prop-erties of chains of prime ideals in integral extensions.The phrase going up refers to the case when a chain can be extended by “upward inclusion", while going down refersto the case when a chain can be extended by “downward inclusion”.The major results are the Cohen–Seidenberg theorems, which were proved by Irvin S. Cohen and Abraham Sei-denberg. These are known as the going-up and going-down theorems.

6.1 Going up and going down

Let A⊆B be an extension of commutative rings.The going-up and going-down theorems give sufficient conditions for a chain of prime ideals in B, each member ofwhich lies over members of a longer chain of prime ideals in A, to be able to be extended to the length of the chainof prime ideals in A.

6.1.1 Lying over and incomparability

First, we fix some terminology. If p and q are prime ideals of A and B, respectively, such that

q ∩A = p

(note that q∩A is automatically a prime ideal of A) then we say that p lies under q and that q lies over p . In general,a ring extension A⊆B of commutative rings is said to satisfy the lying over property if every prime ideal P of A liesunder some prime ideal Q of B.The extension A⊆B is said to satisfy the incomparability property if whenever Q and Q' are distinct primes of Blying over prime P in A, then Q⊈Q' and Q' ⊈Q.

6.1.2 Going-up

The ring extension A⊆B is said to satisfy the going-up property if whenever

p1 ⊆ p2 ⊆ · · · ⊆ pn

is a chain of prime ideals of A and

q1 ⊆ q2 ⊆ · · · ⊆ qm

13

14 CHAPTER 6. GOING UP AND GOING DOWN

(m < n) is a chain of prime ideals of B such that for each 1 ≤ i ≤ m, qi lies over pi , then the latter chain can beextended to a chain

q1 ⊆ q2 ⊆ · · · ⊆ qn

such that for each 1 ≤ i ≤ n, qi lies over pi .In (Kaplansky 1970) it is shown that if an extension A⊆B satisfies the going-up property, then it also satisfies thelying-over property.

6.1.3 Going down

The ring extension A⊆B is said to satisfy the going-down property if whenever

p1 ⊇ p2 ⊇ · · · ⊇ pn

is a chain of prime ideals of A and

q1 ⊇ q2 ⊇ · · · ⊇ qm

(m < n) is a chain of prime ideals of B such that for each 1 ≤ i ≤ m, qi lies over pi , then the latter chain can beextended to a chain

q1 ⊇ q2 ⊇ · · · ⊇ qn

such that for each 1 ≤ i ≤ n, qi lies over pi .There is a generalization of the ring extension case with ring morphisms. Let f : A→ B be a (unital) ring homomor-phism so that B is a ring extension of f(A). Then f is said to satisfy the going-up property if the going-up propertyholds for f(A) in B.Similarly, if f(A) is a ring extension, then f is said to satisfy the going-down property if the going-down propertyholds for f(A) in B.In the case of ordinary ring extensions such as A⊆B, the inclusion map is the pertinent map.

6.2 Going-up and going-down theorems

The usual statements of going-up and going-down theorems refer to a ring extension A⊆B:

1. (Going up) If B is an integral extension of A, then the extension satisfies the going-up property (and hence thelying over property), and the incomparability property.

2. (Going down) If B is an integral extension of A, and B is a domain, and A is integrally closed in its field offractions, then the extension (in addition to going-up, lying-over and incomparability) satisfies the going-downproperty.

There is another sufficient condition for the going-down property:

• If A⊆B is a flat extension of commutative rings, then the going-down property holds.[1]

Proof:[2] Let p1⊆p2 be prime ideals of A and let q2 be a prime ideal of B such that q2 ∩ A = p2. We wish to provethat there is a prime ideal q1 of B contained in q2 such that q1 ∩ A = p1. Since A⊆B is a flat extension of rings, itfollows that Ap2⊆Bq2 is a flat extension of rings. In fact, Ap2⊆Bq2 is a faithfully flat extension of rings since theinclusion map Ap2 → Bq2 is a local homomorphism. Therefore, the induced map on spectra Spec(Bq2) → Spec(Ap2)is surjective and there exists a prime ideal of Bq2 that contracts to the prime ideal p1Ap2 of Ap2. The contractionof this prime ideal of Bq2 to B is a prime ideal q1 of B contained in q2 that contracts to p1. The proof is complete.Q.E.D.

6.3. REFERENCES 15

6.3 References[1] This follows from a much more general lemma in Bruns-Herzog, Lemma A.9 on page 415.

[2] Matsumura, page 33, (5.D), Theorem 4

• Atiyah, M. F., and I. G. Macdonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9 MR 242802

• Winfried Bruns; Jürgen Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, 39.Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1

• Kaplansky, Irving, Commutative rings, Allyn and Bacon, 1970.

• Matsumura, Hideyuki (1970). Commutative algebra. W. A. Benjamin. ISBN 978-0-8053-7025-6.

• Sharp, R. Y. (2000). “13 Integral dependence on subrings (13.38 The going-up theorem, pp. 258–259; 13.41The going down theorem, pp. 261–262)". Steps in commutative algebra. LondonMathematical Society StudentTexts 51 (Second ed.). Cambridge: Cambridge University Press. pp. xii+355. ISBN 0-521-64623-5. MR1817605.

Chapter 7

Ideal (order theory)

In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this termhistorically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to adifferent notion. Ideals are of great importance for many constructions in order and lattice theory.

7.1 Basic definitions

A non-empty subset I of a partially ordered set (P,≤) is an ideal, if the following conditions hold:[1][2]

1. For every x in I, y ≤ x implies that y is in I. (I is a lower set)

2. For every x, y in I, there is some element z in I, such that x ≤ z and y ≤ z. (I is a directed set)

While this is the most general way to define an ideal for arbitrary posets, it was originally defined for lattices only. Inthis case, the following equivalent definition can be given: a subset I of a lattice (P,≤) is an ideal if and only if it is alower set that is closed under finite joins (suprema), i.e., it is nonempty and for all x, y in I, the element x ∨ y of P isalso in I.[3]

The dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging ∨ with ∧ , is a filter.Some authors use the term ideal to mean a lower set, i.e., they include only condition 1 above.[4][5][6] With this weakerdefinition, an ideal of a lattice seen as a poset is not closed under joins, so it is not necessarily an ideal of the lattice.[6]Wikipedia uses only “ideal/filter (of order theory)" and “lower/upper set” to avoid confusion.Frink ideals, pseudoideals and Doyle pseudoideals are different generalizations of the notion of a lattice ideal.An ideal or filter is said to be proper if it is not equal to the whole set P.[3]

The smallest ideal that contains a given element p is a principal ideal and p is said to be a principal element of theideal in this situation. The principal ideal ↓ p for a principal p is thus given by ↓ p = {x in P | x ≤ p}.

7.2 Prime ideals

An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e.ideals in the inverse order. Such ideals are called prime ideals. Also note that, since we require ideals and filters tobe non-empty, every prime ideal is necessarily proper. For lattices, prime ideals can be characterized as follows:A subset I of a lattice (P,≤) is a prime ideal, if and only if

1. I is a proper ideal of P, and

2. for every elements x and y of P, x ∧ y in I implies that x is in I or y is in I.

16

7.3. MAXIMAL IDEALS 17

It is easily checked that this is indeed equivalent to stating that P\I is a filter (which is then also prime, in the dualsense).For a complete lattice the further notion of a completely prime ideal is meaningful. It is defined to be a proper idealI with the additional property that, whenever the meet (infimum) of some arbitrary set A is in I, some element of Ais also in I. So this is just a specific prime ideal that extends the above conditions to infinite meets.The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot bederived within ZF (Zermelo–Fraenkel set theory without the axiom of choice). This issue is discussed in variousprime ideal theorems, which are necessary for many applications that require prime ideals.

7.3 Maximal ideals

An ideal I ismaximal if it is proper and there is no proper ideal J that is a strictly greater set than I. Likewise, a filterF is maximal if it is proper and there is no proper filter that is strictly greater.When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse of thisstatement is false in general.Maximal filters are sometimes called ultrafilters, but this terminology is often reserved for Boolean algebras, wherea maximal filter (ideal) is a filter (ideal) that contains exactly one of the elements {a, ¬a}, for each element a of theBoolean algebra. In Boolean algebras, the terms prime ideal and maximal ideal coincide, as do the terms prime filterand maximal filter.There is another interesting notion of maximality of ideals: Consider an ideal I and a filter F such that I is disjointfrom F. We are interested in an ideal M that is maximal among all ideals that contain I and are disjoint from F. Inthe case of distributive lattices such an M is always a prime ideal. A proof of this statement follows.

Proof. Assume the idealM is maximal with respect to disjointness from the filter F. Supposefor a contradiction that M is not prime, i.e. there exists a pair of elements a and b such thata ∧ b inM but neither a nor b are inM. Consider the case that for all m inM, m ∨ a is not inF. One can construct an ideal N by taking the downward closure of the set of all binary joinsof this form, i.e. N = { x | x≤ m ∨ a for some m inM}. It is readily checked that N is indeedan ideal disjoint from F which is strictly greater than M. But this contradicts the maximalityof M and thus the assumption that M is not prime.For the other case, assume that there is some m in M with m ∨ a in F. Now if any elementn in M is such that n ∨ b is in F, one finds that (m ∨ n) ∨ b and (m ∨ n) ∨ a are both in F.But then their meet is in F and, by distributivity, (m ∨ n) ∨ (a ∧ b) is in F too. On the otherhand, this finite join of elements of M is clearly in M, such that the assumed existence of ncontradicts the disjointness of the two sets. Hence all elements n ofM have a join with b thatis not in F. Consequently one can apply the above construction with b in place of a to obtainan ideal that is strictly greater than M while being disjoint from F. This finishes the proof.

However, in general it is not clear whether there exists any idealM that is maximal in this sense. Yet, if we assume theAxiom of Choice in our set theory, then the existence of M for every disjoint filter–ideal-pair can be shown. In thespecial case that the considered order is a Boolean algebra, this theorem is called the Boolean prime ideal theorem.It is strictly weaker than the Axiom of Choice and it turns out that nothing more is needed for many order theoreticapplications of ideals.

7.4 Applications

The construction of ideals and filters is an important tool in many applications of order theory.

• In Stone’s representation theorem for Boolean algebras, the maximal ideals (or, equivalently via the negationmap, ultrafilters) are used to obtain the set of points of a topological space, whose clopen sets are isomorphicto the original Boolean algebra.

18 CHAPTER 7. IDEAL (ORDER THEORY)

• Order theory knows many completion procedures, to turn posets into posets with additional completenessproperties. For example, the ideal completion of a given partial order P is the set of all ideals of P ordered bysubset inclusion. This construction yields the free dcpo generated by P. An ideal is principal if and only if it iscompact in the ideal completion, so the original poset can be recovered as the sub-poset consisting of compactelements. Furthermore, every algebraic dcpo can be reconstructed as the ideal completion of its set of compactelements.

7.5 History

Ideals were introduced first by Marshall H. Stone, who derived their name from the ring ideals of abstract algebra.He adopted this terminology because, using the isomorphism of the categories of Boolean algebras and of Booleanrings, the two notions do indeed coincide.

7.6 Literature

Ideals and filters are among the most basic concepts of order theory. See the introductory books given for ordertheory and lattice theory, and the literature on the Boolean prime ideal theorem.

7.7 See also• Filter (mathematics)

• Ideal (ring theory)

• Ideal (set theory)

7.8 Notes[1] Taylor (1999), p. 141: “A directed lower subset of a poset X is called an ideal”

[2] Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S. (2003). Continuous Lattices andDomains. Encyclopedia of Mathematics and its Applications 93. Cambridge University Press. p. 3. ISBN 0521803381.

[3] Burris & Sankappanavar 1981, Def. 8.2.

[4] Lawson (1998), p. 22

[5] Stanley (2002), p. 100

[6] Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 20 and 44

7.9 References• Burris, Stanley N.; Sankappanavar, Hanamantagouda P. (1981). A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

• Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. ISBN 978-981-02-3316-7.

• Stanley, R.P. (2002). Enumerative combinatorics. Cambridge studies in advanced mathematics 1. CambridgeUniversity Press. ISBN 978-0-521-66351-9.

• Taylor, Paul (1999), Practical foundations of mathematics, Cambridge Studies in Advanced Mathematics 59,Cambridge University Press, Cambridge, ISBN 0-521-63107-6, MR 1694820

Chapter 8

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. Ideals generalize certain subsetsof the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preservesevenness, and multiplying an even number by any other integer results in another even number; these closure andabsorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarlyto the way that, in group theory, a normal subgroup can be used to construct a quotient group.Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is aprincipal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals maybe distinct from the ring elements, and certain properties of integers, when generalized to rings, attach more naturallyto the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers,and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization forthe ideals of a Dedekind domain (a type of ring important in number theory).The concept of an order ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal isa generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.

8.1 History

Ideals were first proposed byRichardDedekind in 1876 in the third edition of his bookVorlesungen über Zahlentheorie(English: Lectures on Number Theory). They were a generalization of the concept of ideal numbers developed byErnst Kummer.[1][2] Later the concept was expanded by David Hilbert and especially Emmy Noether.

8.2 Definitions

For an arbitrary ring (R,+, ·) , let (R,+) be its additive group. A subset I is called a two-sided ideal (or simply anideal) of R if it is an additive subgroup of R that “absorbs multiplication by elements of R". Formally we mean thatI is an ideal if it satisfies the following conditions:

1. (I,+) is a subgroup of (R,+)

2. ∀x ∈ I, ∀r ∈ R : x · r ∈ I

3. ∀x ∈ I, ∀r ∈ R : r · x ∈ I.

Equivalently, an ideal of R is a sub-R-bimodule of R.A subset I of R is called a right ideal of R [3] if it is an additive subgroup of R and absorbs multiplication on theright, that is:

1. (I,+) is a subgroup of (R,+)

2. ∀x ∈ I, ∀r ∈ R : x · r ∈ I.

19

20 CHAPTER 8. IDEAL (RING THEORY)

Equivalently, a right ideal of R is a right R -submodule of R .Similarly a subset I of R is called a left ideal of R if it is an additive subgroup of R absorbing multiplication on theleft:

1. (I,+) is a subgroup of (R,+)

2. ∀x ∈ I, ∀r ∈ R : r · x ∈ I.

Equivalently, a left ideal of R is a left R -submodule of R .In all cases, the first condition can be replaced by the following well-known criterion that ensures a nonempty subsetof a group is a subgroup:

1. I is non-empty and ∀x, y ∈ I : x− y ∈ I .[4]

The left ideals in R are exactly the right ideals in the opposite ring Ro and vice versa. A two-sided ideal is a left idealthat is also a right ideal, and is often called an ideal except to emphasize that there might exist single-sided ideals.When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is usedalone.

8.3 Properties

{0} and R are ideals in every ring R. If R is a division ring or a field, then these are its only ideals. The ideal R iscalled the unit ideal. I is a proper ideal if it is a proper subset of R, that is, I does not equal R.[5]

Just as normal subgroups of groups are kernels of group homomorphisms, ideals have interpretations as kernels. Fora nonempty subset A of R:

• A is an ideal of R if and only if it is a kernel of a ring homomorphism from R.

• A is a right ideal of R if and only if it is a kernel of a homomorphism from the right R module RR to anotherright R module.

• A is a left ideal of R if and only if it is a kernel of a homomorphism from the left R module RR to another leftR module.

If p is in R, then pR is a right ideal and Rp is a left ideal of R. These are called, respectively, the principal right andleft ideals generated by p. To remember which is which, note that right ideals are stable under right-multiplication(IR ⊆ I) and left ideals are stable under left-multiplication (RI ⊆ I).The connection between cosets and ideals can be seen by switching the operation from “multiplication” to “addition”.

8.4 Motivation

Intuitively, the definition can be motivated as follows: Suppose we have a subset of elements Z of a ring R and thatwe would like to obtain a ring with the same structure as R, except that the elements of Z should be zero (they are insome sense “negligible”).But if z1 = 0 and z2 = 0 in our new ring, then surely z1 + z2 should be zero too, and rz1 as well as z1r should bezero for any element r (zero or not).The definition of an ideal is such that the ideal I generated (see below) by Z is exactly the set of elements that areforced to become zero if Z becomes zero, and the quotient ring R/I is the desired ring where Z is zero, and onlyelements that are forced by Z to be zero are zero. The requirement that R and R/I should have the same structure(except that I becomes zero) is formalized by the condition that the projection from R to R/I is a (surjective) ringhomomorphism.

8.5. EXAMPLES 21

8.5 Examples• In a ring R, the set R itself forms an ideal of R. Also, the subset containing only the additive identity 0R formsan ideal. These two ideals are usually referred to as the trivial ideals of R.

• The even integers form an ideal in the ring Z of all integers; it is usually denoted by 2Z . This is because thesum of any even integers is even, and the product of any integer with an even integer is also even. Similarly,the set of all integers divisible by a fixed integer n is an ideal denoted nZ .

• The set of all polynomials with real coefficients which are divisible by the polynomial x2 + 1 is an ideal in thering of all polynomials.

• The set of all n-by-n matrices whose last row is zero forms a right ideal in the ring of all n-by-n matrices. It isnot a left ideal. The set of all n-by-n matrices whose last column is zero forms a left ideal but not a right ideal.

• The ring C(R) of all continuous functions f from R to R under pointwise multiplication contains the ideal ofall continuous functions f such that f(1) = 0. Another ideal in C(R) is given by those functions which vanishfor large enough arguments, i.e. those continuous functions f for which there exists a number L > 0 such thatf(x) = 0 whenever |x| > L.

• Compact operators form an ideal in the ring of bounded operators.

8.6 Ideal generated by a set

Let R be a (possibly not unital) ring. Any intersection of any nonempty family of left ideals of R is again a left idealof R. If X is any subset of R, then the intersection of all left ideals of R containing X is a left ideal I of R containing X,and is clearly the smallest left ideal to do so. This ideal I is said to be the left ideal generated by X. Similar definitionscan be created by using right ideals or two-sided ideals in place of left ideals.If R has unity, then the left, right, or two-sided ideal of R generated by a subset X of R can be expressed internally aswe will now describe. The following set is a left ideal:

{r1x1 + · · ·+ rnxn | n ∈ N, ri ∈ R, xi ∈ X}.

Each element described would have to be in every left ideal containing X, so this left ideal is in fact the left idealgenerated by X. The right ideal and ideal generated by X can also be expressed in the same way:

{x1r1 + · · ·+ xnrn | n ∈ N, ri ∈ R, xi ∈ X}

{r1x1s1 + · · ·+ rnxnsn | n ∈ N, ri ∈ R, si ∈ R, xi ∈ X}.

The former is the right ideal generated by X, and the latter is the ideal generated by X.By convention, 0 is viewed as the sum of zero such terms, agreeing with the fact that the ideal of R generated by ∅ is{0} by the previous definition.If a left ideal I of R has a finite subset F such that I is the left ideal generated by F, then the left ideal I is said to befinitely generated. Similar terms are also applied to right ideals and two-sided ideals generated by finite subsets.In the special case where the set X is just a singleton {a} for some a in R, then the above definitions turn into thefollowing:

Ra = {ra | r ∈ R}

aR = {ar | r ∈ R}

RaR = {r1as1 + · · ·+ rnasn | n ∈ N, ri ∈ R, si ∈ R}.

These ideals are known as the left/right/two-sided principal ideals generated by a. It is also very common to denotethe two-sided ideal generated by a as (a).

22 CHAPTER 8. IDEAL (RING THEORY)

If R does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sumsof products of things in X with things in R, we must allow the addition of n-fold sums of the form x+x+...+x, andn-fold sums of the form (−x)+(−x)+...+(−x) for every x in X and every n in the natural numbers. When R has a unit,this extra requirement becomes superfluous.

8.6.1 Example

• In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain),and the only two generators of pR are p and −p. The concepts of “ideal” and “number” are therefore almostidentical in Z. If aR = bR in an arbitrary domain, then au = b for some unit u. Conversely, for any unit u,aR = auu−1R = auR. So, in a commutative principal ideal domain, the generators of the ideal aR are just theelements au where u is an arbitrary unit. This explains the case of Z since 1 and −1 are the only units of Z.

8.7 Types of ideals

To simplify the description all rings are assumed to be commutative. The non-commutative case is discussedin detail in the respective articles.

Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings.Different types of ideals are studied because they can be used to construct different types of factor rings.

• Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I aproper subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutativerings.[6]

• Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.

• Prime ideal: A proper ideal I is called a prime ideal if for any a and b in R, if ab is in I, then at least oneof a and b is in I. The factor ring of a prime ideal is a prime ring in general and is an integral domain forcommutative rings.

• Radical ideal or semiprime ideal: A proper ideal I is called radical or semiprime if for any a in R, if an isin I for some n, then a is in I. The factor ring of a radical ideal is a semiprime ring for general rings, and is areduced ring for commutative rings.

• Primary ideal: An ideal I is called a primary ideal if for all a and b in R, if ab is in I, then at least one ofa and bn is in I for some natural number n. Every prime ideal is primary, but not conversely. A semiprimeprimary ideal is prime.

• Principal ideal: An ideal generated by one element.

• Finitely generated ideal: This type of ideal is finitely generated as a module.

• Primitive ideal: A left primitive ideal is the annihilator of a simple left module. A right primitive ideal isdefined similarly. Actually (despite the name) the left and right primitive ideals are always two-sided ideals.Primitive ideals are prime. A factor rings constructed with a right (left) primitive ideals is a right (left) primitivering. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields.

• Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals whichproperly contain it.

• Comaximal ideals: Two ideals i, j are said to be comaximal if x+ y = 1 for some x ∈ i and y ∈ j .

• Regular ideal: This term has multiple uses. See the article for a list.

• Nil ideal: An ideal is a nil ideal if each of its elements is nilpotent.

Two other important terms using “ideal” are not always ideals of their ring. See their respective articles for details:

8.8. FURTHER PROPERTIES 23

• Fractional ideal: This is usually defined when R is a commutative domain with quotient field K. Despite theirnames, fractional ideals are R submodules of K with a special property. If the fractional ideal is containedentirely in R, then it is truly an ideal of R.

• Invertible ideal: Usually an invertible idealA is defined as a fractional ideal for which there is another fractionalideal B such that AB=BA=R. Some authors may also apply “invertible ideal” to ordinary ring ideals A and Bwith AB=BA=R in rings other than domains.

8.8 Further properties• In rings with identity, an ideal is proper if and only if it does not contain 1 or equivalently it does not containa unit.

• The set of ideals of any ring are partially ordered via subset inclusion, in fact they are additionally a completemodular lattice in this order with join operation given by addition of ideals and meet operation given by setintersection. The trivial ideals supply the least and greatest elements: the largest ideal is the entire ring, and thesmallest ideal is the zero ideal. The lattice is not, in general, a distributive lattice.

• Unfortunately Zorn’s lemma does not necessarily apply to the collection of proper ideals of R. However whenR has identity 1, this collection can be reexpressed as “the collection of ideals which do not contain 1”. It canbe checked that Zorn’s lemma now applies to this collection, and consequently there are maximal proper idealsof R. With a little more work, it can be shown that every proper ideal is contained in a maximal ideal. SeeKrull’s theorem at maximal ideal.

• The ring R can be considered as a left module over itself, and the left ideals of R are then seen as the submodulesof this module. Similarly, the right ideals are submodules of R as a right module over itself, and the two-sidedideals are submodules of R as a bimodule over itself. If R is commutative, then all three sorts of module arethe same, just as all three sorts of ideal are the same.

• Every ideal is a pseudo-ring.

• The ideals of a ring form a semiring (with identity element R) under addition and multiplication of ideals.

8.9 Ideal operations

The sum and product of ideals are defined as follows. For a and b , ideals of a ring R,

a+ b := {a+ b | a ∈ a and b ∈ b}

and

ab := {a1b1 + · · ·+ anbn | ai ∈ a and bi ∈ b, i = 1, 2, . . . , n; for n = 1, 2, . . . },

i.e. the product of two ideals a and b is defined to be the ideal ab generated by all products of the form ab with a ina and b in b . The product ab is contained in the intersection of a and b .The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of allideals of a given ring forms a complete modular lattice. Also, the union of two ideals is a subset of the sum of thosetwo ideals, because for any element a inside an ideal, we can write it as a+0, or 0+a, therefore, it is contained in thesum as well. However, the union of two ideals is not necessarily an ideal.

8.10 Ideals and congruence relations

There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect thering structure) on the ring:

24 CHAPTER 8. IDEAL (RING THEORY)

Given an ideal I of a ring R, let x ~ y if x − y ∈ I. Then ~ is a congruence relation on R.Conversely, given a congruence relation ~ on R, let I = {x : x ~ 0}. Then I is an ideal of R.

8.11 See also• Modular arithmetic

• Noether isomorphism theorem

• Boolean prime ideal theorem

• Ideal theory

• Ideal (order theory)

• Ideal quotient

• Ideal norm

• Artinian ideal

• Noncommutative ring

• Regular ideal

• Idealizer

8.12 References[1] Harold M. Edwards (1977). Fermat’s last theorem. A genetic introduction to algebraic number theory. p. 76.

[2] Everest G., Ward T. (2005). An introduction to number theory. p. 83.

[3] See Hazewinkel et al. (2004), p. 4.

[4] In fact, since R is assumed to be unital, it suffices that x + y is in I, since the second condition implies that −y is in I.

[5] Lang 2005, Section III.2

[6] Because simple commutative rings are fields. See Lam (2001). A First Course in Noncommutative Rings. p. 39.

• Lang, Serge (2005). Undergraduate Algebra (Third ed.). Springer-Verlag. ISBN 978-0-387-22025-3

• Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras,rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0

Chapter 9

Ideal class group

In mathematics, for a field K an ideal class group (or class group) is the quotient group JK/PK where JK is thewhole fractional ideals of K and PK is the principal ideals of K. The extent to which unique factorization fails in thering of integers of an algebraic number field (or more generally any Dedekind domain) can be described by the idealclass group (or class group). If this group is finite (as it is in the case of the ring of integers of a number field), thenthe order of the group is called the class number. The multiplicative theory of a Dedekind domain is intimately tiedto the structure of its class group. For example, the class group of a Dedekind domain is trivial if and only if the ringis a unique factorization domain.

9.1 History and origin of the ideal class group

Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of anideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadraticforms, as put into something like a final form by Gauss, a composition law was defined on certain equivalence classesof forms. This gave a finite abelian group, as was recognised at the time.Later Kummer was working towards a theory of cyclotomic fields. It had been realised (probably by several people)that failure to complete proofs in the general case of Fermat’s last theorem by factorisation using the roots of unitywas for a very good reason: a failure of the fundamental theorem of arithmetic to hold in the rings generated by thoseroots of unity was a major obstacle. Out of Kummer’s work for the first time came a study of the obstruction to thefactorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion in thatgroup for the field of p-roots of unity, for any prime number p, as the reason for the failure of the standard methodof attack on the Fermat problem (see regular prime).Somewhat later again Dedekind formulated the concept of ideal, Kummer having worked in a different way. At thispoint the existing examples could be unified. It was shown that while rings of algebraic integers do not always haveunique factorization into primes (because they need not be principal ideal domains), they do have the property thatevery proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integersis a Dedekind domain). The size of the ideal class group can be considered as a measure for the deviation of a ringfrom being a principal domain; a ring is a principal domain if and only if it has a trivial ideal class group.

9.2 Definition

If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzeroelements a and b of R such that (a)I = (b)J. (Here the notation (a) means the principal ideal of R consisting of allthe multiples of a.) It is easily shown that this is an equivalence relation. The equivalence classes are called the idealclasses of R. Ideal classes can be multiplied: if [I] denotes the equivalence class of the ideal I, then the multiplication[I][J] = [IJ] is well-defined and commutative. The principal ideals form the ideal class [R] which serves as an identityelement for this multiplication. Thus a class [I] has an inverse [J] if and only if there is an ideal J such that IJ is aprincipal ideal. In general, such a J may not exist and consequently the set of ideal classes of Rmay only be a monoid.However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain,

25

26 CHAPTER 9. IDEAL CLASS GROUP

the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class groupof R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain,every non-zero ideal (except R) is a product of prime ideals.

9.3 Properties

The ideal class group is trivial (i.e. has only one element) if and only if all ideals of R are principal. In this sense, theideal class group measures how far R is from being a principal ideal domain, and hence from satisfying unique primefactorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains).The number of ideal classes (the class number of R) may be infinite in general. In fact, every abelian group isisomorphic to the ideal class group of some Dedekind domain.[1] But if R is in fact a ring of algebraic integers, thenthe class number is always finite. This is one of the main results of classical algebraic number theory.Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an algebraicnumber field of small discriminant, using Minkowski’s bound. This result gives a bound, depending on the ring, suchthat every ideal class contains an ideal norm less than the bound. In general the bound is not sharp enough to makethe calculation practical for fields with large discriminant, but computers are well suited to the task.The mapping from rings of integers R to their corresponding class groups is functorial, and the class group canbe subsumed under the heading of algebraic K-theory, with K0(R) being the functor assigning to R its ideal classgroup; more precisely, K0(R) = Z×C(R), where C(R) is the class group. Higher K groups can also be employed andinterpreted arithmetically in connection to rings of integers.

9.4 Relation with the group of units

It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in aDedekind domain behave like elements. The other part of the answer is provided by the multiplicative group of unitsof the Dedekind domain, since passage from principal ideals to their generators requires the use of units (and this isthe rest of the reason for introducing the concept of fractional ideal, as well):Define a map from K× to the set of all nonzero fractional ideals of R by sending every element to the principal(fractional) ideal it generates. This is a group homomorphism; its kernel is the group of units of R, and its cokernelis the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map to be anisomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.

9.5 Examples of ideal class groups

• The rings Z, Z[ω], and Z[i], where ω is a cube root of 1 and i is a fourth root of 1 (i.e. a square root of −1),are all principal ideal domains (and in fact are all Euclidean domains), and so have class number 1: that is, theyhave trivial ideal class groups.

• If k is a field, then the polynomial ring k[X1, X2, X3, ...] is an integral domain. It has a countably infinite setof ideal classes.

9.5.1 Class numbers of quadratic fields

If d is a square-free integer (a product of distinct primes) other than 1, then Q(√d) is a quadratic extension of Q.If d < 0, then the class number of the ring R of algebraic integers of Q(√d) is equal to 1 for precisely the followingvalues of d: d = −1, −2, −3, −7, −11, −19, −43, −67, and −163. This result was first conjectured by Gauss andproven by Kurt Heegner, although Heegner’s proof was not believed until Harold Stark gave a later proof in 1967.(See Stark-Heegner theorem.) This is a special case of the famous class number problem.If, on the other hand, d > 0, then it is unknown whether there are infinitely many fields Q(√d) with class number1. Computational results indicate that there are a great many such fields. However, it is not even known if there areinfinitely many number fields with class number 1.[2][3]

9.6. CONNECTIONS TO CLASS FIELD THEORY 27

For d < 0, the ideal class group of Q(√d) is isomorphic to the class group of integral binary quadratic forms ofdiscriminant equal to the discriminant of Q(√d). For d > 0, the ideal class group may be half the size since the classgroup of integral binary quadratic forms is isomorphic to the narrow class group of Q(√d).[4]

Example of a non-trivial class group

The quadratic integer ring R = Z [√−5] is the ring of integers of Q(√−5). It does not possess unique factorization; infact the class group of R is cyclic of order 2. Indeed, the ideal

J = (2, 1 + √−5)

is not principal, which can be proved by contradiction as follows. R has a norm functionN(a+ b√−5) = a2 +5b2

, which satisfies N(uv) = N(u)N(v) , and N(u) = 1 if and only if u is a unit in R . First of all, J = R , becausethe quotient ring of R modulo the ideal (1 +

√−5) is isomorphic to Z/6Z , so that the quotient ring of R modulo

J is isomorphic to Z/2Z . If J were generated by an element x of R, then x would divide both 2 and 1 + √−5. Thenthe norm N(x) would divide both N(2) = 4 and N(1 +

√5) = 6 , so N(x) would divide 2. If N(x) = 1 , then x

is a unit, and J = R , a contradiction. But N(x) cannot be 2 either, because R has no elements of norm 2, becausethe Diophantine equation b2 + 5c2 = 2 has no solutions in integers, as it has no solutions modulo 5.One also computes that J2 = (2), which is principal, so the class of J in the ideal class group has order two. Showingthat there aren't any other ideal classes requires more effort.The fact that this J is not principal is also related to the fact that the element 6 has two distinct factorisations intoirreducibles:

6 = 2 × 3 = (1 + √−5) × (1 − √−5).

9.6 Connections to class field theory

Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a givenalgebraic number field, meaning Galois extensions with abelian Galois group. A particularly beautiful example isfound in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extensionof such a field. The Hilbert class field L of a number field K is unique and has the following properties:

• Every ideal of the ring of integers of K becomes principal in L, i.e., if I is an integral ideal of K then the imageof I is a principal ideal in L.

• L is a Galois extension of K with Galois group isomorphic to the ideal class group of K.

Neither property is particularly easy to prove.

9.7 See also• Class number formula

• Class number problem

• Brauer–Siegel theorem—an asymptotic formula for the class number

• List of number fields with class number one

• Principal ideal domain

• Algebraic K-theory

• Galois theory

• Fermat’s last theorem

28 CHAPTER 9. IDEAL CLASS GROUP

• Narrow class group

• Picard group—a generalisation of the class group appearing in algebraic geometry

• Arakelov class group

9.8 Notes[1] Claborn 1966

[2] Neukirch 1999

[3] Gauss 1700

[4] Fröhlich & Taylor 1993, Theorem 58

9.9 References• Claborn, Luther (1966), “Every abelian group is a class group”, Pacific Journal of Mathematics 18: 219–222,doi:10.2140/pjm.1966.18.219

• Fröhlich, Albrecht; Taylor, Martin (1993), Algebraic number theory, Cambridge Studies in Advanced Mathe-matics 27, Cambridge University Press, ISBN 978-0-521-43834-6, MR 1215934

• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322,Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859

Chapter 10

Ideal norm

In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It isparticularly important in number theory since it measures the size of an ideal of a complicated number ring in termsof an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z,then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.

10.1 Relative norm

Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K.(this implies that B is also a Dedekind domain.) Let IA and IB be the ideal groups of A and B, respectively (i.e., thesets of nonzero fractional ideals.) Following (Serre 1979), the norm map

NB/A : IB → IA

is the unique group homomorphism that satisfies

NB/A(q) = p[B/q:A/p]

for all nonzero prime ideals q of B, where p = q ∩A is the prime ideal of A lying below q .Alternatively, for any b ∈ IB one can equivalently define NB/A(b) to be the fractional ideal of A generated by theset {NL/K(x)|x ∈ b} of field norms of elements of B.[1]

For a ∈ IA , one hasNB/A(aB) = an , where n = [L : K] . The ideal norm of a principal ideal is thus compatiblewith the field norm of an element: NB/A(xB) = NL/K(x)A. [2]

Let L/K be a Galois extension of number fields with rings of integers OK ⊂ OL . Then the preceding applies withA = OK , B = OL , and for any b ∈ IOL

we have

NOL/OK(b) = OK ∩

∏σ∈Gal(L/K)

σ(b),

which is an element of IOK. The notation NOL/OK

is sometimes shortened to NL/K , an abuse of notation that iscompatible with also writing NL/K for the field norm, as noted above.In the case K = Q , it is reasonable to use positive rational numbers as the range for NOL/Z since Z has trivialideal class group and unit group {±1} , thus each nonzero fractional ideal of Z is generated by a uniquely determinedpositive rational number. Under this convention the relative norm fromL down toK = Q coincides with the absolutenorm defined below.

29

30 CHAPTER 10. IDEAL NORM

10.2 Absolute norm

Let L be a number field with ring of integersOL , and a a nonzero (integral) ideal ofOL . The absolute norm of a is

N(a) := [OL : a] = |OL/a|.

By convention, the norm of the zero ideal is taken to be zero.If a = (a) is a principal ideal, then N(a) = |NL/Q(a)| .[3]

The norm is completely multiplicative: if a and b are ideals ofOL , thenN(a ·b) = N(a)N(b) .[4] Thus the absolutenorm extends uniquely to a group homomorphism

N : IOL → Q×>0,

defined for all nonzero fractional ideals of OL .The norm of an ideal a can be used to give an upper bound on the field norm of the smallest nonzero element itcontains: there always exists a nonzero a ∈ a for which

|NL/Q(a)| ≤(2

π

)s √|∆L|N(a),

where ∆L is the discriminant of L and s is the number of pairs of (non-real) complex embeddings of L into C (thenumber of complex places of L ).[5]

10.3 See also• Field norm

• Dedekind zeta function

10.4 References[1] Janusz, Proposition I.8.2

[2] Serre, 1.5, Proposition 14.

[3] Marcus, Theorem 22c, pp. 65-66.

[4] Marcus, Theorem 22a, pp. 65-66

[5] Neukirch, Lemma 6.2

• Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies inMathematics 7 (second ed.), Providence,Rhode Island: American Mathematical Society, pp. x+276, ISBN 0-8218-0429-4, MR 1362545 (96j:11137)

• Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, pp. viii+279, ISBN 0-387-90279-1, MR 0457396 (56 #15601)

• Jürgen Neukirch (1999), Algebraic number theory, Berlin: Springer-Verlag, pp. xviii+571, ISBN 3-540-65399-6, MR 1697859 (2000m:11104)

• Serre, Jean-Pierre (1979), Local fields, Graduate Texts inMathematics 67, Translated from the French byMar-vin Jay Greenberg, New York: Springer-Verlag, pp. viii+241, ISBN 0-387-90424-7, MR 554237 (82e:12016)

Chapter 11

Ideal quotient

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set

(I : J) = {r ∈ R|rJ ⊂ I}

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because IJ ⊂ K if and only ifI ⊂ K : J . The ideal quotient is useful for calculating primary decompositions. It also arises in the description ofthe set difference in algebraic geometry (see below).(I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is arelated notion of the inverse of a fractional ideal.

11.1 Properties

The ideal quotient satisfies the following properties:

• (I : J) = AnnR((J + I)/I) asR -modules, where AnnR(M) denotes the annihilator ofM as anR -module.

• J ⊂ I ⇒ I : J = R

• I : R = I

• R : I = R

• I : (J +K) = (I : J) ∩ (I : K)

• I : (r) = 1r (I ∩ (r)) (as long as R is an integral domain)

11.2 Calculating the quotient

The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. Forexample, if I = (f1, f2, f3) and J = (g1, g2) are ideals in k[x1, ..., xn], then

I : J = (I : (g1)) ∩ (I : (g2)) =

(1

g1(I ∩ (g1))

)∩(

1

g2(I ∩ (g2))

)Then elimination theory can be used to calculate the intersection of I with (g1) and (g2):

I ∩ (g1) = tI + (1− t)(g1) ∩ k[x1, . . . , xn], I ∩ (g2) = tI + (1− t)(g1) ∩ k[x1, . . . , xn]

Calculate a Gröbner basis for tI + (1-t)(g1) with respect to lexicographic order. Then the basis functions which haveno t in them generate I ∩ (g1) .

31

32 CHAPTER 11. IDEAL QUOTIENT

11.3 Geometric interpretation

The ideal quotient corresponds to set difference in algebraic geometry.[1] More precisely,

• IfW is an affine variety and V is a subset of the affine space (not necessarily a variety), then

I(V ) : I(W ) = I(V \W )

where I(•) denotes the taking of the ideal associated to a subset.

• If I and J are ideals in k[x1, ..., xn], with k algebraically closed and I radical then

Z(I : J) = cl(Z(I) \ Z(J))

where cl(•) denotes the Zariski closure, and Z(•) denotes the taking of the variety defined by an ideal. If I is notradical, then the same property holds if we saturate the ideal J:

Z(I : J∞) = cl(Z(I) \ Z(J))

where J∞ = J + J2 + · · ·+ Jn + · · · .

11.4 References[1] David Cox, John Little, and Donal O'Shea (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational

Algebraic Geometry and Commutative Algebra. Springer. ISBN 0-387-94680-2., p.195

Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol130 (AMS 2012)M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.

Chapter 12

Ideal theory

12.1 In mathematics

In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contem-porary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker–Noethertheorem in algebraic geometry, and the ideal class group in algebraic number theory, of the commutative algebra ofthe first quarter of the twentieth century. It was used in the influential van der Waerden text on abstract algebra fromaround 1930.The ideal theory in question had been based on elimination theory, but in line with David Hilbert's taste moved awayfrom algorithmic methods. Gröbner basis theory has now reversed the trend, for computer algebra.The importance of the ideal in general of a module, more general than an ideal, probably led to the perception thatideal theory was too narrow a description. Valuation theory, too, was an important technical extension, and was usedby Helmut Hasse and Oscar Zariski. Bourbaki used commutative algebra; sometimes local algebra is applied to thetheory of local rings. D. G. Northcott's 1953 Cambridge Tract Ideal Theory (reissued 2004 under the same title) wasone of the final appearances of the name.

12.2 In political philosophy

In political philosophy, ideal theory refers to argument concerning political or social arrangements under favorableassumptions. The phrase is associated with the work of John Rawls.[1]

12.3 References[1] Stanford Encyclopedia of Philosophy, article on Rawls

33

Chapter 13

Jacobian ideal

In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or functiongerm. LetO(x1, . . . , xn) denote the ring of smooth functions and f a function in the ring. The Jacobian ideal of f is

Jf :=

⟨∂f

∂x1, . . . ,

∂f

∂xn

⟩.

13.1 See also• Milnor number

• Unfolding

34

Chapter 14

Jacobson radical

In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is theideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting “left” inplace of “right” in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radicalof a ring is frequently denoted by J(R) or rad(R); however to avoid confusion with other radicals of rings, the formernotation will be preferred in this article. The Jacobson radical is named after Nathan Jacobson, who was the first tostudy it for arbitrary rings in (Jacobson 1945).The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfullyextend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical toinclude modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such asNakayama’s lemma.

14.1 Intuitive discussion

As with other radicals of rings, the Jacobson radical can be thought of as a collection of “bad” elements. In thiscase the “bad” property is that these elements annihilate all simple left and right modules of the ring. For purposes ofcomparison, consider the nilradical of a commutative ring, which consists of all elements that are nilpotent. In factfor any ring, the nilpotent elements in the center of the ring are also in the Jacobson radical.[1] So, for commutativerings, the nilradical is contained in the Jacobson radical.The Jacobson radical is very similar to the nilradical in an intuitive sense. A weaker notion of being bad, weaker thanbeing a zero divisor, is being a non-unit (not invertible under multiplication). The Jacobson radical of a ring consistsof elements that satisfy a stronger property than being merely a non-unit – in some sense, a member of the Jacobsonradical must not “act as a unit” in anymodule “internal to the ring.” More precisely, a member of the Jacobson radicalmust project under the canonical homomorphism to the zero of every “right division ring” (each non-zero elementof which has a right inverse) internal to the ring in question. Concisely, it must belong to every maximal right idealof the ring. These notions are of course imprecise, but at least explain why the nilradical of a commutative ring iscontained in the ring’s Jacobson radical.In yet a simpler way, we may think of the Jacobson radical of a ring as method to “mod out bad elements” of the ring –that is, members of the Jacobson radical act as 0 in the quotient ring, R/J(R). IfN is the nilradical of commutative ringR, then the quotient ring R/N has no nilpotent elements. Similarly for any ring R, the quotient ring has J(R/J(R))={0}and so all of the “bad” elements in the Jacobson radical have been removed by modding out J(R). Elements of theJacobson radical and nilradical can be therefore seen as generalizations of 0.

14.2 Equivalent characterizations

The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appearin many noncommutative algebra texts such as (Anderson 1992, §15), (Isaacs 1993, §13B), and (Lam 2001, Ch 2).The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for ringswithout unity are given immediately afterward):

35

36 CHAPTER 14. JACOBSON RADICAL

• J(R) equals the intersection of all maximal right ideals of the ring. It is also true that J(R) equals the intersectionof all maximal left ideals within the ring.[2] These characterizations are internal to the ring, since one only needsto find the maximal right ideals of the ring. For example, if a ring is local, and has a unique maximal rightideal, then this unique maximal right ideal is an ideal because it is exactly J(R). Maximal ideals are in a senseeasier to look for than annihilators of modules. This characterization is deficient, however, because it doesnot prove useful when working computationally with J(R). The left-right symmetry of these two definitionsis remarkable and has various interesting consequences.[3][2] This symmetry stands in contrast to the lack ofsymmetry in the socles of R, for it may happen that soc(RR) is not equal to soc(RR). If R is a non-commutativering, J(R) is not necessarily equal to the intersection of all maximal two-sided ideals of R. For instance, if V isa countable direct sum of copies of a field k and R=End(V) (the ring of endomorphisms of V as a k-module),then J(R)=0 because R is known to be von Neumann regular, but there is exactly one maximal double-sidedideal in R consisting of endomorphisms with finite-dimensional image. (Lam 2001, p. 46, Ex. 3.15)

• J(R) equals the sum of all superfluous right ideals (or symmetrically, the sum of all superfluous left ideals) ofR. Comparing this with the previous definition, the sum of superfluous right ideals equals the intersection ofmaximal right ideals. This phenomenon is reflected dually for the right socle of R: soc(RR) is both the sum ofminimal right ideals and the intersection of essential right ideals. In fact, these two relationships hold for theradicals and socles of modules in general.

• As defined in the introduction, J(R) equals the intersection of all annihilators of simple right R-modules, how-ever it is also true that it is the intersection of annihilators of simple left modules. An ideal that is the annihilatorof a simple module is known as a primitive ideal, and so a reformulation of this states that the Jacobson radicalis the intersection of all primitive ideals. This characterization is useful when studying modules over rings. Forinstance, if U is right R-module, and V is a maximal submodule of U, U·J(R) is contained in V, where U·J(R)denotes all products of elements of J(R) (the “scalars”) with elements in U, on the right. This follows from thefact that the quotient module, U/V is simple and hence annihilated by J(R).

• J(R) is the unique right ideal of R maximal with the property that every element is right quasiregular.[4][1]Alternatively, one could replace “right” with “left” in the previous sentence.[2] This characterization of theJacobson radical is useful both computationally and in aiding intuition. Furthermore, this characterization isuseful in studying modules over a ring. Nakayama’s lemma is perhaps the most well-known instance of this.Although every element of the J(R) is necessarily quasiregular, not every quasiregular element is necessarily amember of J(R).[1]

• While not every quasiregular element is in J(R), it can be shown that y is in J(R) if and only if xy is leftquasiregular for all x in R. (Lam 2001, p. 50)

• J(R) is the set of all such elements x ∈ R that every element of 1 + RxR is a unit: J(R) = {x ∈ R |1 +RxR ⊂ R× } .

For rings without unity it is possible for R=J(R), however the equation that J(R/J(R))={0} still holds. The followingare equivalent characterizations of J(R) for rings without unity appear in (Lam 2001, p. 63):

• The notion of left quasiregularity can be generalized in the following way. Call an element a inR left generalizedquasiregular if there exists c in R such that c+a-ca= 0. Then J(R) consists of every element a for which rais left generalized quasiregular for all r in R. It can be checked that this definition coincides with the previousquasiregular definition for rings with unity.

• For a ring without unity, the definition of a left simplemoduleM is amended by adding the condition thatR•M ≠0. With this understanding, J(R) may be defined as the intersection of all annihilators of simple left Rmodules,or just R if there are no simple left R modules. Rings without unity with no simple modules do exist, in whichcase R=J(R), and the ring is called a radical ring. By using the generalized quasiregular characterization ofthe radical, it is clear that if one finds a ring with J(R) nonzero, then J(R) is a radical ring when considered asa ring without unity.

14.3. EXAMPLES 37

14.3 Examples

• Rings for which J(R) is {0} are called semiprimitive rings, or sometimes “Jacobson semisimple rings”. TheJacobson radical of any field, any von Neumann regular ring and any left or right primitive ring is {0}. TheJacobson radical of the integers is {0}.

• The Jacobson radical of the ring Z/12Z (see modular arithmetic) is 6Z/12Z, which is the intersection of themaximal ideals 2Z/12Z and 3Z/12Z.

• If K is a field and R is the ring of all upper triangular n-by-n matrices with entries in K, then J(R) consists ofall upper triangular matrices with zeros on the main diagonal.

• If K is a field and R = K[[X1, ..., Xn]] is a ring of formal power series, then J(R) consists of those power serieswhose constant term is zero. More generally: the Jacobson radical of every local ring is the unique maximalideal of the ring.

• Start with a finite, acyclic quiver Γ and a field K and consider the quiver algebra KΓ (as described in the quiverarticle). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1.

• The Jacobson radical of a C*-algebra is {0}. This follows from the Gelfand–Naimark theorem and the factfor a C*-algebra, a topologically irreducible *-representation on a Hilbert space is algebraically irreducible, sothat its kernel is a primitive ideal in the purely algebraic sense (see spectrum of a C*-algebra).

14.4 Properties

• If R is unital and is not the trivial ring {0}, the Jacobson radical is always distinct from R since rings with unityalways have maximal right ideals. However, some important theorems and conjectures in ring theory considerthe case when J(R) = R - “If R is a nil ring (that is, each of its elements is nilpotent), is the polynomial ringR[x] equal to its Jacobson radical?" is equivalent to the open Köthe conjecture. (Smoktunowicz 2006, p. 260,§5)

• The Jacobson radical of the ring R/J(R) is zero. Rings with zero Jacobson radical are called semiprimitiverings.

• A ring is semisimple if and only if it is Artinian and its Jacobson radical is zero.

• If f : R→ S is a surjective ring homomorphism, then f(J(R)) ⊆ J(S).

• If M is a finitely generated left R-module with J(R)M = M, then M = 0 (Nakayama’s lemma).

• J(R) contains all central nilpotent elements, but contains no idempotent elements except for 0.

• J(R) contains every nil ideal of R. If R is left or right Artinian, then J(R) is a nilpotent ideal. This can actuallybe made stronger: If {0} = T0 ⊆ T1 ⊆ · · · ⊆ Tk = R is a composition series for the right R-module R (sucha series is sure to exist if R is right artinian, and there is a similar left composition series if R is left artinian),then (J (R))

k= 0 . (Proof: Since the factors Tu/Tu−1 are simple right R-modules, right multiplication by any

element of J(R) annihilates these factors. In other words, (Tu/Tu−1) ·J (R) = 0 , whence Tu ·J (R) ⊆ Tu−1

. Consequently, induction over i shows that all nonnegative integers i and u (for which the following makessense) satisfy Tu ·(J (R))

i ⊆ Tu−i . Applying this to u = i = k yields the result.) Note, however, that in generalthe Jacobson radical need not consist of only the nilpotent elements of the ring.

• If R is commutative and finitely generated as an algebra over either a field or Z, then J(R) is equal to thenilradical of R.

• The Jacobson radical of a (unital) ring is its largest superfluous right (equivalently, left) ideal.

38 CHAPTER 14. JACOBSON RADICAL

14.5 See also• Nilradical

• Radical of a module

• Radical of an ideal

• Frattini subgroup

14.6 Notes[1] Isaacs, p. 181.

[2] Isaacs, p. 182.

[3] Isaacs, Problem 12.5, p. 173

[4] Isaacs, Corollary 13.4, p. 180

14.7 References• Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics13 (2 ed.), New York: Springer-Verlag, pp. x+376, ISBN 0-387-97845-3, MR 1245487 (94i:16001)

• Atiyah, M. F.; Macdonald, I. G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co.,Reading, Mass.-London-Don Mills, Ont., pp. ix+128, MR 0242802 (39 #4129)

• N. Bourbaki. Éléments de Mathématique.

• Herstein, I. N. (1994) [1968], Noncommutative rings, Carus Mathematical Monographs 15, Washington, DC:Mathematical Association ofAmerica, pp. xii+202, ISBN0-88385-015-X,MR1449137 (97m:16001) Reprintof the 1968 original; With an afterword by Lance W. Small

• Isaacs, I. M. (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.

• Jacobson, Nathan (1945), “The radical and semi-simplicity for arbitrary rings”, American Journal of Mathe-matics 67: 300–320, doi:10.2307/2371731, ISSN 0002-9327, MR 12271

• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), NewYork: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439 (2002c:16001)

• Pierce, Richard S. (1982), Associative algebras, Graduate Texts in Mathematics 88, New York: Springer-Verlag, pp. xii+436, ISBN 0-387-90693-2, MR674652 (84c:16001) Studies in theHistory ofModern Science,9

• Smoktunowicz, Agata (2006), “Some results in noncommutative ring theory”, International Congress of Math-ematicians, Vol. II (PDF), European Mathematical Society, pp. 259–269, ISBN 978-3-03719-022-7, MR2275597

Chapter 15

Krull’s principal ideal theorem

In commutative algebra,Krull’s principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a boundon the height of a principal ideal in a Noetherian ring. The theorem is sometimes referred to by its German name,Krulls Hauptidealsatz (Satz meaning “proposition” or “theorem”).Formally, if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.This theorem can be generalized to ideals that are not principal, and the result is often calledKrull’s height theorem.This says that if R is a Noetherian ring and I is a proper ideal generated by n elements of R, then I has height at mostn.The principal ideal theorem and the generalization, the height theorem, both follow from the fundamental theoremof dimension theory. Bourbaki’s Commutative Algebra gives a direct proof. Kaplansky’s Commutative ring includesa proof due to David Rees.

15.1 References• Matsumura, Hideyuki (1970), Commutative Algebra, New York: Benjamin, see in particular section (12.I), p.77

• http://www.math.lsa.umich.edu/~{}hochster/615W10/supDim.pdf

39

Chapter 16

Krull’s theorem

In mathematics, and more specifically in ring theory, Krull’s theorem, named after Wolfgang Krull, asserts thata nonzero ring[1] has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfiniteinduction. The theorem admits a simple proof using Zorn’s lemma, and in fact is equivalent to Zorn’s lemma, whichin turn is equivalent to the axiom of choice.

16.1 Variants

• For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold.

• For pseudo-rings, the theorem holds for regular ideals.

• A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows:

Let R be a ring, and let I be a proper ideal of R. Then there is a maximal ideal of Rcontaining I.

This result implies the original theorem, by taking I to be the zero ideal (0). Conversely, applying theoriginal theorem to R/I leads to this result.To prove the stronger result directly, consider the set S of all proper ideals of R containing I. The set Sis nonempty since I ∈ S. Furthermore, for any chain T of S, the union of the ideals in T is an ideal J,and a union of ideals not containing 1 does not contain 1, so J ∈ S. By Zorn’s lemma, S has a maximalelement M. This M is a maximal ideal containing I.

16.2 Krull’s Hauptidealsatz

Main article: Krull’s principal ideal theorem

Another theorem commonly referred to as Krull’s theorem:

Let R be a Noetherian ring and a an element of R which is neither a zero divisornor a unit. Then every minimal prime ideal P containing a has height 1.

16.3 Notes

[1] In this article, rings have a 1.

40

16.4. REFERENCES 41

16.4 References• W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathematische Annalen 10 (1929), 729–744.

Chapter 17

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to setinclusion) amongst all proper ideals.[1][2] In other words, I is a maximal ideal of a ring R if there are no other idealscontained between I and R.Maximal ideals are important because the quotient rings of maximal ideals are simple rings, and in the special caseof unital commutative rings they are also fields.In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in theposet of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset ofproper left ideals. Since a one sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarilya ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, andthe maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in factthe Jacobson radical J(R).It is possible for a ring to have a unique maximal ideal and yet lack unique maximal one sided ideals: for example,in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal ideal, but there are many maximal rightideals.

17.1 Definition

There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals.Given a ring R and a proper ideal I of R (that is I ≠ R), I is a maximal ideal of R if any of the following equivalentconditions hold:

• There exists no other proper ideal J of R so that I ⊊ J.

• For any ideal J with I ⊆ J, either J = I or J = R.

• The quotient ring R/I is a simple ring.

There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right idealA of a ring R, the following conditions are equivalent to A being a maximal right ideal of R:

• There exists no other proper right ideal B of R so that A ⊊ B.

• For any right ideal B with A ⊆ B, either B = A or B = R.

• The quotient module R/A is a simple right R module.

Maximal right/left/two-sided ideals are the dual notion to that of minimal ideals.

42

17.2. EXAMPLES 43

17.2 Examples

• If F is a field, then the only maximal ideal is {0}.

• In the ring Z of integers, the maximal ideals are the principal ideals generated by a prime number.

• More generally, all nonzero prime ideals are maximal in a principal ideal domain.

• The maximal ideals of the polynomial ring K[x1,...,xn] over an algebraically closed field K are the ideals of theform (x1 − a1,...,xn − an). This result is known as the weak nullstellensatz.

17.3 Properties

• An important ideal of the ring called the Jacobson radical can be defined using maximal right (or maximal left)ideals.

• If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal.In that case, R/m is known as the residue field. This fact can fail in non-unital rings. For example, 4Z is amaximal ideal in 2Z , but 2Z/4Z is not a field.

• If L is a maximal left ideal, then R/L is a simple left R module. Conversely in rings with unity, any simple leftR module arises this way. Incidentally this shows that a collection of representatives of simple left R modulesis actually a set since it can be put into correspondence with part of the set of maximal left ideals of R.

• Krull’s theorem (1929): Every nonzero ring with a multiplicative identity has a maximal ideal. The result isalso true if “ideal” is replaced with “right ideal” or “left ideal”. More generally, it is true that every nonzerofinitely generated module has a maximal submodule. Suppose I is an ideal which is not R (respectively, A is aright ideal which is not R). Then R/I is a ring with unity, (respectively, R/A is a finitely generated module), andso the above theorems can be applied to the quotient to conclude that there is a maximal ideal (respectivelymaximal right ideal) of R containing I (respectively, A).

• Krull’s theorem can fail for rings without unity. A radical ring, i.e. a ring in which the Jacobson radical is theentire ring, has no simple modules and hence has no maximal right or left ideals. See regular ideals for possibleways to circumvent this problem.

• In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true: forexample, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal. Commutativerings in which prime ideals are maximal are known as zero-dimensional rings, where the dimension used is theKrull dimension.

17.4 Generalization

For an R module A, a maximal submodule M of A is a submodule M≠A for which for any other submodule N, ifM⊆N⊆A then N=M or N=A. Equivalently, M is a maximal submodule if and only if the quotient module A/M is asimple module. Clearly the maximal right ideals of a ring R are exactly the maximal submodules of the module RR.Unlike rings with unity however, a module does not necessarily have maximal submodules. However, as noted above,finitely generated nonzeromodules havemaximal submodules, and also projectivemodules havemaximal submodules.As with rings, one can define the radical of a module using maximal submodules.Furthermore, maximal ideals can be generalized by defining a maximal sub-bimodule M of a bimodule B to be aproper sub-bimodule of M which is contained by no other proper sub-bimodule of M. So, the maximal ideals of Rare exactly the maximal sub-bimodules of the bimodule RRR.

44 CHAPTER 17. MAXIMAL IDEAL

17.5 References[1] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.

[2] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

• Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics13 (2 ed.), New York: Springer-Verlag, pp. x+376, ISBN 0-387-97845-3, MR 1245487

• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), NewYork: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439

Chapter 18

Minimal ideal

In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a nonzero right idealwhich contains no other nonzero right ideal. Likewise aminimal left ideal is a nonzero left ideal of R containing noother nonzero left ideals of R, and a minimal ideal of R is a nonzero ideal containing no other nonzero two-sidedideal of R. (Isaacs 2009, p.190)Said another way, minimal right ideals are minimal elements of the poset of nonzero right ideals of R ordered byinclusion. The reader is cautioned that outside of this context, some posets of ideals may admit the zero ideal, andso zero could potentially be a minimal element in that poset. This is the case for the poset of prime ideals of a ring,which may include the zero ideal as a minimal prime ideal.

18.1 Definition

The definition of a minimal right ideal N of a module R is equivalent to the following conditions:

• If K is a right ideal of R with {0} ⊆ K ⊆ N, then either K = {0} or K = N.

• N is a simple right R module.

Minimal right ideals are the dual notion to the idea of maximal right ideals.

18.2 Properties

Many standard facts on minimal ideals can be found in standard texts such as (Anderson & Fuller 1999), (Isaacs1992), (Lam 2001), and (Lam 1999).

• It is a fact that in a ring with unity, maximal right ideals always exist. In contrast, there is no guarantee thatminimal right, left, or two-sided ideals exist in a ring.

• The right socle of a ring soc(RR) is an important structure defined in terms of the minimal right ideals of R.

• Rings for which every right ideal contains a minimal right ideal are exactly the rings with an essential rightsocle.

• Any right Artinian ring or right Kasch ring has a minimal right ideal.

• Domains which are not division rings have no minimal right ideals.

• In rings with unity, minimal right ideals are necessarily principal right ideals, because for any nonzero x in aminimal right ideal N, the set xR is a nonzero right ideal of R inside N, and so xR=N.

• Brauer’s lemma: Any minimal right ideal N in a ring R satisfies N2={0} or N=eR for some idempotentelement of R. (Lam 2001, p.162)

45

46 CHAPTER 18. MINIMAL IDEAL

• If N1 and N2 are nonisomorphic minimal right ideals of R, then the product N1N2={0}.

• If N1 and N2 are distinct minimal ideals of a ring R, then N1N2={0}.

• A simple ring with a minimal right ideal is a semisimple ring.

• In a semiprime ring, there exists a minimal right ideal if and only if there exists a minimal left ideal. (Lam2001, p.174)

18.3 Generalization

A nonzero submodule N of a right module M is called a minimal submodule if it contains no other nonzero sub-modules of M. Equivalently, N is a nonzero submodule of M which is a simple module. This can also be extendedto bimodules by calling a nonzero sub-bimodule N a minimal sub-bimodule of M if N contains no other nonzerosub-bimodules.If the module M is taken to be the right R module RR, then clearly the minimal submodules are exactly the minimalright ideals of R. Likewise, the minimal left ideals of R are precisely the minimal submodules of the left module RR.In the case of two-sided ideals, we see that the minimal ideals of R are exactly the minimal sub-bimodules of thebimodule RRR.Just as with rings, there is no guarantee that minimal submodules exist in a module. Minimal submodules can be usedto define the socle of a module.

18.4 References• Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics13 (2 ed.), New York: Springer-Verlag, pp. x+376, ISBN 0-387-97845-3, MR 1245487

• Isaacs, I. Martin (2009) [1994], Algebra: a graduate course, Graduate Studies inMathematics 100, Providence,RI: American Mathematical Society, pp. xii+516, ISBN 978-0-8218-4799-2, MR 2472787

• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), NewYork: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439

18.5 External links

http://www.encyclopediaofmath.org/index.php/Minimal_ideal

Chapter 19

Minimal prime ideal

In mathematics, especially in the area of algebra known as commutative algebra, certain prime ideals calledminimalprime ideals play an important role in understanding rings and modules. The notion of height and Krull’s principalideal theorem use minimal primes.

19.1 Definition

A prime ideal P is said to be aminimal prime ideal over an ideal I if it is minimal among all prime ideals containingI. (Note that we do not exclude I even if it is a prime ideal.) A prime ideal is said to be a minimal prime ideal if itis a minimal prime ideal over the zero ideal.A minimal prime ideal over an ideal I in a Noetherian ring R is precisely a minimal associated prime (also calledisolated prime) of R/I ; this follows for instance from the primary decomposition of I.

19.2 Examples• In a commutative artinian ring, every maximal ideal is a minimal prime ideal.

• In an integral domain, the only minimal prime ideal is the zero ideal.

• In the ring Z of integers, the minimal prime ideals over a nonzero principal ideal (n) are the principal ideals(p), where p is a prime divisor of n. The only minimal prime ideal over the zero ideal is the zero ideal itself.Similar statements hold for any principal ideal domain.

• If I is a p-primary ideal (for example, a power of p), then p is the unique minimal prime ideal over I.

19.3 Properties

All rings are assumed to be commutative and unital.

• Every proper ideal I in a ring has at least one minimal prime ideal above it. The proof of this fact uses Zorn’slemma (Kaplansky 1974, p. 6). Any maximal ideal containing I is prime, and such ideals exist, so the setof prime ideals containing I is non-empty. The intersection of a decreasing chain of prime ideals is prime.Therefore, the set of prime ideals containing I has a minimal element, which is a minimal prime over I.

• Emmy Noether showed that in a Noetherian ring, there are only finitely many minimal prime ideals over anygiven ideal. (Kaplansky 1974, p. 59), (Eisenbud 1995, p. 47) The fact remains true if “Noetherian” is replacedby the ascending chain conditions on radical ideals.

• The radical√I of any proper ideal I coincides with the intersection of the minimal prime ideals over I.

(Kaplansky 1974, p. 16).

47

48 CHAPTER 19. MINIMAL PRIME IDEAL

• The set of zero divisors of a given ring contains the union of the minimal prime ideals (Kaplansky 1974, p.57).

• Krull’s principal ideal theorem describes important properties of minimal prime ideals.

19.4 References• Eisenbud, David (1995), Commutative algebra, Graduate Texts inMathematics 150, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960

• Kaplansky, Irving (1974), Commutative rings, University of Chicago Press, MR 0345945

Chapter 20

Nil ideal

In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if eachof its elements is nilpotent.[1][2]

The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respectto the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutativerings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture.

20.1 Commutative rings

In a commutative ring, the set of all nilpotent elements forms an ideal known as the nilradical of the ring. Therefore,an ideal of a commutative ring is nil if, and only if, it is a subset of the nilradical; that is, the nilradical is the idealmaximal with respect to the property that each of its elements is nilpotent.In commutative rings, the nil ideals are more well-understood compared to the case of noncommutative rings. This isprimarily because the commutativity assumption ensures that the product of two nilpotent elements is again nilpotent.For instance, if a is a nilpotent element of a commutative ring R, a·R is an ideal that is in fact nil. This is because anyelement of the principal ideal generated by a is of the form a·r for r in R, and if an = 0, (a·r)n = an·rn = 0. It is notin general true however, that a·R is a nil (one-sided) ideal in a noncommutative ring, even if a is nilpotent.

20.2 Noncommutative rings

The theory of nil ideals is of major importance in noncommutative ring theory. In particular, through the under-standing of nil rings—rings whose every element is nilpotent—one may obtain a much better understanding of moregeneral rings.[3]

In the case of commutative rings, there is always a maximal nil ideal: the nilradical of the ring. The existence of sucha maximal nil ideal in the case of noncommutative rings is guaranteed by the fact that the sum of nil ideals is againnil. However, the truth of the assertion that the sum of two left nil ideals is again a left nil ideal remains elusive; itis an open problem known as the Köthe conjecture.[4] The Köthe conjecture was first posed in 1930 and yet remainsunresolved as of 2010.

20.3 Relation to nilpotent ideals

The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the twonotions coincide. If an ideal is nilpotent, it is of course nil. There are two main barriers for nil ideals to be nilpotent:

1. There need not be an upper bound on the exponent required to annihilate elements. Arbitrarily high exponentsmay be required.

2. The product of n nilpotent elements may be nonzero for arbitrarily high n.

49

50 CHAPTER 20. NIL IDEAL

Clearly both of these barriers must be avoided for a nil ideal to qualify as nilpotent.In a right artinian ring, any nil ideal is nilpotent.[5] This is proven by observing that any nil ideal is contained in theJacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the artinian hypothesis), theresult follows. In fact, this has been generalized to right noetherian rings; the result is known as Levitzky’s theorem.A particularly simple proof due to Utumi can be found in (Herstein 1968, Theorem 1.4.5, p. 37).

20.4 See also• Köthe conjecture

• Nilpotent ideal

• Nilradical

• Jacobson radical

20.5 Notes[1] Isaacs 1993, p. 194

[2] Herstein 1968, Definition (b), p. 13

[3] Section 2 of Smoktunowicz 2006, p. 260

[4] Herstein 1968, p. 21

[5] Isaacs, Corollary 14.3, p. 195.

20.6 References• Herstein, I. N. (1968), Noncommutative rings (1st ed.), The Mathematical Association of America, ISBN 0-88385-015-X

• Isaacs, I. Martin (1993), Algebra, a graduate course (1st ed.), Brooks/Cole Publishing Company, ISBN 0-534-19002-2

• Smoktunowicz, Agata (2006), “Some results in noncommutative ring theory” (PDF), International Congress ofMathematicians, Vol. II, Zürich: European Mathematical Society, pp. 259–269, ISBN 978-3-03719-022-7,MR 2275597, retrieved 2009-08-19

Chapter 21

Nilpotent ideal

In mathematics, more specifically ring theory, an ideal, I, of a ring is said to be a nilpotent ideal, if there exists anatural number k such that Ik = 0.[1] By Ik, it is meant the additive subgroup generated by the set of all products ofk elements in I.[1] Therefore, I is nilpotent if and only if there is a natural number k such that the product of any kelements of I is 0.The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however,instances when the two notions coincide—this is exemplified by Levitzky’s theorem.[2][3] The notion of a nilpotentideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.

21.1 Relation to nil ideals

The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the twonotions coincide. If an ideal is nilpotent, it is of course nil, but a nil ideal need not be nilpotent for more than onereason. The first is that there need not be a global upper bound on the exponent required to annihilate various elementsof the nil ideal, and secondly, each element being nilpotent does not force products of distinct elements to vanish.[1]

In a right artinian ring, any nil ideal is nilpotent.[4] This is proven by observing that any nil ideal is contained in theJacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the artinian hypothesis), theresult follows. In fact, this can be generalized to right noetherian rings; this result is known as Levitzky’s theorem.[3]

21.2 See also

• Köthe conjecture

• Nilpotent element

• Nil ideal

• Nilradical

• Jacobson radical

21.3 Notes[1] Isaacs, p. 194.

[2] Isaacs, Theorem 14.38, p. 210

[3] Herstein, Theorem 1.4.5, p. 37.

[4] Isaacs, Corollary 14.3, p. 195

51

52 CHAPTER 21. NILPOTENT IDEAL

21.4 References• I.N. Herstein (1968). Noncommutative rings (1st edition ed.). The Mathematical Association of America.ISBN 0-88385-015-X.

• I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company. ISBN0-534-19002-2.

Chapter 22

Nilradical of a ring

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring.In the non-commutative ring case the same definition does not always work. This has resulted in several radicalsgeneralizing the commutative case in distinct ways. See the article "radical of a ring" for more of this.The nilradical of a Lie algebra is similarly defined for Lie algebras.

22.1 Commutative rings

The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of thezero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent (by the binomial formula), andthe product of any element with a nilpotent element (by commutativity) is nilpotent. It can also be characterized asthe intersection of all the prime ideals of the ring. (In fact, it is the intersection of all minimal prime ideals.)A ring is called reduced if it has no nonzero nilpotent. Thus, a ring is reduced if and only if its nilradical is zero. IfR is an arbitrary commutative ring, then the quotient of it by the nilradical is a reduced ring and is denoted by Rred .Since every maximal ideal is a prime ideal, the Jacobson radical — which is the intersection of maximal ideals —must contain the nilradical. A ring is called a Jacobson ring if the nilradical of R/P coincides with the Jacobsonradical of R/P for every prime ideal P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotentideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.

22.2 Noncommutative rings

Further information: Radical of a ring

For noncommutative rings, there are several analogues of the nilradical. The lower nilradical (or Baer–McCoy radical,or prime radical) is the analogue of the radical of the zero ideal and is defined as the intersection of the prime ideals ofthe ring. The analogue of the set of all nilpotent elements is the upper nilradical and is defined as the ideal generatedby all nil ideals of the ring, which is itself a nil ideal. The set of all nilpotent elements itself need not be an ideal (oreven a subgroup), so the upper nilradical can be much smaller than this set. The Levitzki radical is in between andis defined as the largest locally nilpotent ideal. As in the commutative case, when the ring is artinian, the Levitzkiradical is nilpotent and so is the unique largest nilpotent ideal. Indeed, if the ring is merely noetherian, then the lower,upper, and Levitzki radical are nilpotent and coincide, allowing the nilradical of any noetherian ring to be defined asthe unique largest (left, right, or two-sided) nilpotent ideal of the ring.

22.3 References• Eisenbud, David, “Commutative Algebra with a View Toward Algebraic Geometry”, Graduate Texts in Math-ematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.

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54 CHAPTER 22. NILRADICAL OF A RING

• Lam, Tsit-Yuen (2001),AFirst Course in Noncommutative Rings (2nd ed.), Berlin, NewYork: Springer-Verlag,ISBN 978-0-387-95325-0, MR 1838439

Chapter 23

Primary ideal

In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primaryif whenever xy is an element of Q then x or yn is also an element of Q, for some n>0. For example, in the ring ofintegers Z, (pn) is a primary ideal if p is a prime number.The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has aprimary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is knownas the Lasker–Noether theorem. Consequently,[1] an irreducible ideal of a Noetherian ring is primary.Various methods of generalizing primary ideals to noncommutative rings exist[2] but the topic is most often studiedfor commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.

23.1 Examples and properties• The definition can be rephrased in a more symmetric manner: an ideal q is primary if, whenever xy ∈ q , wehave either x ∈ q or y ∈ q or x, y ∈ √

q . (Here√q denotes the radical of q .)

• An ideal Q of R is primary if and only if every zerodivisor in R/Q is nilpotent. (Compare this to the case ofprime ideals, where P is prime if every zerodivisor in R/P is actually zero.)

• Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime.

• Every primary ideal is primal.[3]

• If Q is a primary ideal, then the radical of Q is necessarily a prime ideal P, and this ideal is called the associatedprime ideal of Q. In this situation, Q is said to be P-primary.

• On the other hand, an ideal whose radical is prime is not necessarily primary: for example, if R =k[x, y, z]/(xy − z2) , p = (x, z) , and q = p2 , then p is prime and √q = p , but we have xy = z2 ∈p2 = q , x ∈ q , and yn ∈ q for all n > 0, so q is not primary. The primary decomposition of q is(x) ∩ (x2, xz, y) ; here (x) is p -primary and (x2, xz, y) is (x, y, z) -primary.

• An ideal whose radical is maximal, however, is primary.

• If P is a maximal prime ideal, then any ideal containing a power of P is P-primary. Not all P-primary idealsneed be powers of P; for example the ideal (x, y2) is P-primary for the ideal P = (x, y) in the ring k[x, y], butis not a power of P.

• In general powers of a prime ideal P need not be P-primary. (An example is given by taking R to be the ringk[x, y, z]/(xy − z2), with P the prime ideal (x, z). If Q = P2, then xy ∈ Q, but x is not in Q and y is not inthe radical P of Q, so Q is not P-primary.) However every ideal Q with radical P is contained in a smallestP-primary ideal, consisting of all elements a such that ax is in Q for some x not in P. In particular there is asmallest P-primary ideal containing Pn, called the nth symbolic power of P.

55

56 CHAPTER 23. PRIMARY IDEAL

• If A is a Noetherian ring and P a prime ideal, then the kernel of A → AP , the map from A to the localizationof A at P, is the intersection of all P-primary ideals.[4]

23.2 Footnotes[1] To be precise, one usually uses this fact to prove the theorem.

[2] See the references to Chatters-Hajarnavis, Goldman, Gorton-Heatherly, and Lesieur-Croisot.

[3] For the proof of the second part see the article of Fuchs

[4] Atiyah-Macdonald, Corollary 10.21

23.3 References• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, p.50, ISBN 978-0-201-40751-8

• Chatters, A. W.; Hajarnavis, C. R. (1971), “Non-commutative rings with primary decomposition”, Quart. J.Math. Oxford Ser. (2) 22: 73–83, doi:10.1093/qmath/22.1.73, ISSN 0033-5606, MR 0286822

• Goldman, Oscar (1969), “Rings andmodules of quotients”, J. Algebra 13: 10–47, doi:10.1016/0021-8693(69)90004-0, ISSN 0021-8693, MR 0245608

• Gorton, Christine; Heatherly, Henry (2006), “Generalized primary rings and ideals”, Math. Pannon. 17 (1):17–28, ISSN 0865-2090, MR 2215638

• On primal ideals, Ladislas Fuchs

• Lesieur, L.; Croisot, R. (1963), Algèbre noethérienne non commutative (in French), Mémor. Sci. Math., Fasc.CLIV. Gauthier-Villars & Cie, Editeur -Imprimeur-Libraire, Paris, p. 119, MR 0155861

23.4 External links• Primary ideal at Encyclopaedia of Mathematics

Chapter 24

Prime ideal

This article is about ideals in ring theory. For prime ideals in order theory, see ideal (order theory)#Prime ideals.In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring

A Hasse diagram of a portion of the lattice of ideals of the integers Z. The purple nodes indicate prime ideals. The purple and greennodes are semiprime ideals, and the purple and blue nodes are primary ideals.

of integers.[1][2] The prime ideals for the integers are the sets that contain all the multiples of a given prime number,together with the zero ideal.Primitive ideals are prime, and prime ideals are both primary and semiprime.

24.1 Prime ideals for commutative rings

An ideal P of a commutative ring R is prime if it has the following two properties:

• If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P,

• P is not equal to the whole ring R.

This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab oftwo integers, then p divides a or p divides b. We can therefore say

A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z.

57

58 CHAPTER 24. PRIME IDEAL

24.1.1 Examples

• If R denotes the ringC[X,Y] of polynomials in two variables with complex coefficients, then the ideal generatedby the polynomial Y 2 − X 3 − X − 1 is a prime ideal (see elliptic curve).

• In the ring Z[X] of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. Itconsists of all those polynomials whose constant coefficient is even.

• In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M iscontained in exactly two ideals of R, namely M itself and the entire ring R. Every maximal ideal is in factprime. In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general.

• If M is a smooth manifold, R is the ring of smooth real functions on M, and x is a point in M, then the set ofall smooth functions f with f (x) = 0 forms a prime ideal (even a maximal ideal) in R.

24.1.2 Properties

• An ideal I in the ring R (with unity) is prime if and only if the factor ringR/I is an integral domain. In particular,a commutative ring is an integral domain if and only if {0} is a prime ideal.

• An ideal I is prime if and only if its set-theoretic complement is multiplicatively closed.[3]

• Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal), which is adirect consequence of Krull’s theorem.

• The set of all prime ideals (the spectrum of a ring) contains minimal elements (called minimal prime). Geo-metrically, these correspond to irreducible components of the spectrum.

• The preimage of a prime ideal under a ring homomorphism is a prime ideal.

• The sum of two prime ideals is not necessarily prime. For an example, consider the ring C[x, y] with primeideals P = (x2 + y2 − 1) and Q = (x) (the ideals generated by x2 + y2 − 1 and x respectively). Their sum P + Q= (x2 + y2 − 1, x) = (y2 − 1, x) however is not prime: y2 − 1 = (y − 1)(y + 1) ∈ P + Q but its two factors arenot. Alternatively, note that the quotient ring has zero divisors so it is not an integral domain and thus P + Qcannot be prime.

• In a commutative ring R with at least two elements, if every proper ideal is prime, then the ring is a field. (Ifthe ideal (0) is prime, then the ring R is an integral domain. If q is any non-zero element of R and the ideal(q2) is prime, then it contains q and then q is invertible.)

• A nonzero principal ideal is prime if and only if it is generated by a prime element. In a UFD, every nonzeroprime ideal contains a prime element.

24.1.3 Uses

One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in poly-nomial rings. It turns out that the irreducible varieties correspond to prime ideals. In the modern abstract approach,one starts with an arbitrary commutative ring and turns the set of its prime ideals, also called its spectrum, into atopological space and can thus define generalizations of varieties called schemes, which find applications not only ingeometry, but also in number theory.The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that theimportant property of unique factorisation expressed in the fundamental theorem of arithmetic does not hold in everyring of algebraic integers, but a substitute was found when Richard Dedekind replaced elements by ideals and primeelements by prime ideals; see Dedekind domain.

24.2. PRIME IDEALS FOR NONCOMMUTATIVE RINGS 59

24.2 Prime ideals for noncommutative rings

The notion of a prime ideal can be generalized to noncommutative rings by using the commutative definition “ideal-wise”. Wolfgang Krull advanced this idea in 1928.[4] The following content can be found in texts such as (Goodearl2004) and (Lam 2001). If R is a (possibly noncommutative) ring and P is an ideal in R other than R itself, we saythat P is prime if for any two ideals A and B of R:

• If the product of ideals AB is contained in P, then at least one of A and B is contained in P.

It can be shown that this definition is equivalent to the commutative one in commutative rings. It is readily verifiedthat if an ideal of a noncommutative ring R satisfies the commutative definition of prime, then it also satisfies thenoncommutative version. An ideal P satisfying the commutative definition of prime is sometimes called a completelyprime ideal to distinguish it from other merely prime ideals in the ring. Completely prime ideals are prime ideals,but the converse is not true. For example, the zero ideal in the ring of n × n matrices over a field is a prime ideal, butit is not completely prime.This is close to the historical point of view of ideals as ideal numbers, as for the ring Z “A is contained in P” is anotherway of saying “P divides A”, and the unit ideal R represents unity.Equivalent formulations of the ideal P ≠ R being prime include the following properties:

• For all a and b in R, (a)(b) ⊆ P implies a ∈ P or b ∈ P.

• For any two right ideals of R, AB ⊆ P implies A ⊆ P or B ⊆ P.

• For any two left ideals of R, AB ⊆ P implies A ⊆ P or B ⊆ P.

• For any elements a and b of R, if aRb ⊆ P, then a ∈ P or b ∈ P.

Prime ideals in commutative rings are characterized by having multiplicatively closed complements in R, and withslight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. Anonempty subset S ⊆ R is called an m-system if for any a and b in S, there exists r in R such that arb is in S.[5]The following item can then be added to the list of equivalent conditions above:

• The complement R\P is an m-system.

24.2.1 Examples

• Any primitive ideal is prime.

• As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals.

• A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and onlyif the zero ideal is a completely prime ideal.

• Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R module,and P is a maximal element in the poset of annihilator ideals of submodules of A, then P is prime.

24.3 Important facts• Prime avoidance lemma. If R is a commutative ring, and A is a subring (possibly without unity), and I1, ...,In is a collection of ideals of R with at most two members not prime, then if A is not contained in any Ij, it isalso not contained in the union of I1, ..., In.[6] In particular, A could be an ideal of R.

• If S is any m-system in R, then a lemma essentially due to Krull shows that there exists an ideal of R maximalwith respect to being disjoint from S, and moreover the ideal must be prime.[7] In the case {S} = {1}, we haveKrull’s theorem, and this recovers the maximal ideals of R. Another prototypical m-system is the set, {x, x2,x3, x4, ...}, of all positive powers of a non-nilpotent element.

60 CHAPTER 24. PRIME IDEAL

• For a prime ideal P, the complement R\P has another property beyond being an m-system. If xy is in R\P,then both x and y must be in R\P, since P is an ideal. A set that contains the divisors of its elements is calledsaturated.

• For a commutative ring R, there is a kind of converse for the previous statement: If S is any nonempty saturatedand multiplicatively closed subset of R, the complement R\S is a union of prime ideals of R.[8]

• The intersection of members of a descending chain of prime ideals is a prime ideal, and in a commutativering the union of members of an ascending chain of prime ideals is a prime ideal. With Zorn’s Lemma, theseobservations imply that the poset of prime ideals of a commutative ring (partially ordered by inclusion) hasmaximal and minimal elements.

24.4 Connection to maximality

Prime ideals can frequently be produced as maximal elements of certain collections of ideals. For example:

• An ideal maximal with respect to having empty intersection with a fixed m-system is prime.

• An ideal maximal among annihilators of submodules of a fixed R module M is prime.

• In a commutative ring, an ideal maximal with respect to being non-principal is prime.[9]

• In a commutative ring, an ideal maximal with respect to being not countably generated is prime.[10]

24.5 References[1] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.

[2] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

[3] Reid, Miles (1996). Undergraduate Commutative Algebra. Cambridge University Press. ISBN 0-521-45889-7.

[4] Krull, Wolfgang, Primidealketten in allgemeinen Ringbereichen, Sitzungsberichte Heidelberg. Akad. Wissenschaft (1928),7. Abhandl.,3-14.

[5] Obviously, multiplicatively closed sets are m-systems.

[6] Jacobson Basic Algebra II, p. 390

[7] Lam First Course in Noncommutative Rings, p. 156

[8] Kaplansky Commutative rings, p. 2

[9] Kaplansky Commutative rings, p. 10, Ex 10.

[10] Kaplansky Commutative rings, p. 10, Ex 11.

24.6 Further reading

• Goodearl, K. R.; Warfield, R. B., Jr. (2004), An introduction to noncommutative Noetherian rings, LondonMathematical Society Student Texts 61 (2 ed.), Cambridge: Cambridge University Press, pp. xxiv+344, ISBN0-521-54537-4, MR 2080008

• Jacobson, Nathan (1989), Basic algebra. II (2 ed.), New York: W. H. Freeman and Company, pp. xviii+686,ISBN 0-7167-1933-9, MR 1009787

• Kaplansky, Irving (1970), Commutative rings, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, MR 0254021

24.6. FURTHER READING 61

• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2nd ed.), NewYork: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439, Zbl 0980.16001

• Lam, T. Y.; Reyes, Manuel L. (2008), “A prime ideal principle in commutative algebra”, J. Algebra 319 (7):3006–3027, doi:10.1016/j.jalgebra.2007.07.016, ISSN 0021-8693, MR 2397420, Zbl 1168.13002

• Hazewinkel, Michiel, ed. (2001), “Prime ideal”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 25

Primitive ideal

Not to be confused with primary ideal.

In mathematics, a left primitive ideal in ring theory is the annihilator of a simple left module. A right primitive idealis defined similarly. Note that (despite the name) left and right primitive ideals are always two-sided ideals.The quotient of a ring by a left primitive ideal is a left primitive ring.

25.1 References• Isaacs, I. Martin (1994), Algebra, Brooks/Cole Publishing Company, ISBN 0-534-19002-2

62

Chapter 26

Principal ideal

A principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by everyelement of R.

26.1 Definitions

• a left principal ideal of R is a subset of R of the form Ra = {ra : r in R};

• a right principal ideal is a subset of the form aR = {ar : r in R};

• a two-sided principal ideal is a subset of the form RaR = {r1as1 + ... + rnasn : r1,s1,...,rn,sn in R}.

If R is a commutative ring, then the above three notions are all the same. In that case, it is common to write the idealgenerated by a as ⟨a⟩.

26.2 Examples of non-principal ideal

Not all ideals are principal. For example, consider the commutative ring C[x,y] of all polynomials in two variables xand y, with complex coefficients. The ideal ⟨x,y⟩ generated by x and y, which consists of all the polynomials in C[x,y]that have zero for the constant term, is not principal. To see this, suppose that p were a generator for ⟨x,y⟩; then xand y would both be divisible by p, which is impossible unless p is a nonzero constant. But zero is the only constantin ⟨x,y⟩, so we have a contradiction.In the ring Z[\sqrt{−3}] = {a + b\sqrt{−3}: a, b in Z}, numbers in which a + b is even form a non-principal ideal.This ideal forms a regular hexagonal lattice in the complex plane. Consider (a,b) = (2,0) and (1,1). These numbersare elements of this ideal with the same norm (2), but because the only units in the ring are 1 and −1, they are notassociates.

26.3 Related definitions

A ring in which every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID)is an integral domain that is principal. Any PID must be a unique factorization domain; the normal proof of uniquefactorization in the integers (the so-called fundamental theorem of arithmetic) holds in any PID.

26.4 Properties

Any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find agenerator of any ideal. More generally, any two principal ideals in a commutative ring have a greatest common

63

64 CHAPTER 26. PRINCIPAL IDEAL

divisor in the sense of ideal multiplication. In principal ideal domains, this allows us to calculate greatest commondivisors of elements of the ring, up to multiplication by a unit; we define gcd(a,b) to be any generator of the ideal⟨a,b⟩.For a Dedekind domain R, we may also ask, given a non-principal ideal I of R, whether there is some extension S ofR such that the ideal of S generated by I is principal (said more loosely, I becomes principal in S). This question arosein connection with the study of rings of algebraic integers (which are examples of Dedekind domains) in numbertheory, and led to the development of class field theory by Teiji Takagi, Emil Artin, David Hilbert, and many others.The principal ideal theorem of class field theory states that every integer ring R (i.e. the ring of integers of somenumber field) is contained in a larger integer ring S which has the property that every ideal of R becomes a principalideal of S. In this theorem we may take S to be the ring of integers of the Hilbert class field of R; that is, the maximalunramified abelian extension (that is, Galois extension whose Galois group is abelian) of the fraction field of R, andthis is uniquely determined by R.Krull’s principal ideal theorem states that if R is a Noetherian ring and I is a principal, proper ideal of R, then I hasheight at most one.

26.5 See also• Ascending chain condition for principal ideals

26.6 References• Joseph A. Gallian (2004). Contemporary Abstract Algebra. Houghton Mifflin. p. 262. ISBN 978-0-618-51471-7.

Chapter 27

Principal ideal theorem

This article is about the Hauptidealsatz of class field theory. For the theorem about local rings, see Krull’s principalideal theorem.

In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory says thatextending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbertclass field, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been calledprincipalization, or sometimes capitulation.

27.1 Formal statement

For any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, then

IOL

is a principal ideal αOL, for OL the ring of integers of L and some element α in it.

27.2 History

The principal ideal theorem was conjectured by David Hilbert (1902), and was the last remaining aspect of hisprogram on class fields to be completed, in 1929.Emil Artin (1927, 1929) reduced the principal ideal theorem to a question about finite abelian groups: he showedthat it would follow if the transfer from a finite group to its derived subgroup is trivial. This result was proved byPhilipp Furtwängler (1929).

27.3 References• Artin, Emil (1927), “Beweis des allgemeinen Reziprozitätsgesetzes”, Abhandlungen aus dem MathematischenSeminar der Universität Hamburg 5 (1): 353–363, doi:10.1007/BF02952531

• Artin, Emil (1929), “Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz”, Abhandlungen ausdem Mathematischen Seminar der Universität Hamburg 7 (1): 46–51, doi:10.1007/BF02941159

• Furtwängler, Philipp (1929). “Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper”. Abh.Math. Sem. Hamburg 7: 14–36. doi:10.1007/BF02941157. JFM 55.0699.02.

• Gras, Georges (2003). Class field theory. From theory to practice. Springer Monographs in Mathematics.Berlin: Springer-Verlag. ISBN 3-540-44133-6. Zbl 1019.11032.

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66 CHAPTER 27. PRINCIPAL IDEAL THEOREM

• Hilbert, David (1902) [1898], "Über die Theorie der relativ-Abel’schen Zahlkörper”, Acta Mathematica 26(1): 99–131, doi:10.1007/BF02415486

• Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 104. ISBN 3-540-63003-1. Zbl 0819.11044.

• Serre, Jean-Pierre (1979). Local fields. Graduate Texts in Mathematics 67. Translated from the French byMarvin Jay Greenberg. Springer-Verlag. pp. 120–122. ISBN 0-387-90424-7. Zbl 0423.12016.

Chapter 28

Radical of a ring

In ring theory, a branch of mathematics, a radical of a ring is an ideal of “bad” elements of the ring.The first example of a radical was the nilradical introduced in (Köthe 1930), based on a suggestion in (Wedderburn1908). In the next few years several other radicals were discovered, of which the most important example is theJacobson radical. The general theory of radicals was defined independently by (Amitsur 1952, 1954, 1954b) andKurosh (1953).

28.1 Definitions

In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not havean identity element. In particular, every ideal in a ring is also a ring.A radical class (also called radical property or just radical) is a class σ of rings possibly without identities, suchthat:(1) the homomorphic image of a ring in σ is also in σ(2) every ring R contains an ideal S(R) in σ which contains every other ideal in σ(3) S(R/S(R)) = 0. The ideal S(R) is called the radical, or σ-radical, of R.The study of such radicals is called torsion theory.For any class δ of rings, there is a smallest radical class Lδ containing it, called the lower radical of δ. The operatorL is called the lower radical operator.A class of rings is called regular if every non-zero ideal of a ring in the class has a non-zero image in the class. Forevery regular class δ of rings, there is a largest radical classUδ, called the upper radical of δ, having zero intersectionwith δ. The operator U is called the upper radical operator.A class of rings is called hereditary if every ideal of a ring in the class also belongs to the class.

28.2 Examples

28.2.1 The Jacobson radical

Main article: Jacobson radical

Let R be any ring, not necessarily commutative. The Jacobson radical of R is the intersection of the annihilators ofall simple right R-modules.There are several equivalent characterizations of the Jacobson radical, such as:

• J(R) is the intersection of the regular maximal right (or left) ideals of R.

67

68 CHAPTER 28. RADICAL OF A RING

• J(R) is the intersection of all the right (or left) primitive ideals of R.

• J(R) is the maximal right (or left) quasi-regular right (resp. left) ideal of R.

As with the nilradical, we can extend this definition to arbitrary two-sided ideals I by defining J(I) to be the preimageof J(R/I) under the projection map R→R/I.If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra,then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equalto the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.

28.2.2 The Baer radical

The Baer radical of a ring is the intersection of the prime ideals of the ring R. Equivalently it is the smallest semiprimeideal in R. The Baer radical is the lower radical of the class of nilpotent rings. Also called the “lower nilradical” (anddenoted Nil∗R), the “prime radical”, and the “Baer-McCoy radical”. Every element of the Baer radical is nilpotent,so it is a nil ideal.For commutative rings, this is just the nilradical and closely follows the definition of the radical of an ideal.

28.2.3 The upper nil radical or Köthe radical

The sum of the nil ideals of a ring R is the upper nilradical Nil*R or Köthe radical and is the unique largest nil idealof R. Köthe’s conjecture asks whether any left nil ideal is in the nilradical.

28.2.4 Singular radical

An element of a (possibly non-commutative ring) is called left singular if it annihilates an essential left ideal,that is, r is left singular if Ir = 0 for some essential left ideal I. The set of left singular elements of a ring R is atwo-sided ideal, called the left singular ideal, and is denoted Z(RR) . The ideal N of R such that N/Z(RR) =Z(R/Z(RR)R/Z(RR)) is denoted by Z2(RR) and is called the singular radical or the Goldie torsion of R. Thesingular radical contains the prime radical (the nilradical in the case of commutative rings) but may properly containit, even in the commutative case. However, the singular radical of a Noetherian ring is always nilpotent.

28.2.5 The Levitzki radical

The Levitzki radical is defined as the largest locally nilpotent ideal, analogous to the Hirsch–Plotkin radical in thetheory of groups. If the ring is noetherian, then the Levitzki radical is itself a nilpotent ideal, and so is the uniquelargest left, right, or two-sided nilpotent ideal.

28.2.6 The Brown–McCoy radical

The Brown–McCoy radical (called the strong radical in the theory of Banach algebra) can be defined in any of thefollowing ways:

• the intersection of the maximal two-sided ideals

• the intersection of all maximal modular ideals

• the upper radical of the class of all simple rings with identity

The Brown–McCoy radical is studied in much greater generality than associative rings with 1.

28.3. SEE ALSO 69

28.2.7 The von Neumann regular radical

A von Neumann regular ring is a ring A (possibly non-commutative without identity) such that for every a there issome b with a = aba. The von Neumann regular rings form a radical class. It contains every matrix ring over adivision algebra, but contains no nil rings.

28.2.8 The Artinian radical

The Artinian radical is usually defined for two-sided Noetherian rings as the sum of all right ideals that are Artinianmodules. The definition is left-right symmetric, and indeed produces a two-sided ideal of the ring. This radical isimportant in the study of Noetherian rings, as outlined in (Chatters 1980).

28.3 See also

Related uses of radical that are not radicals of rings:

• Radical of a module

• Kaplansky radical

• Radical of a bilinear form

28.4 References• Andrunakievich, V.A. (2001), “Radical of ring and algebras”, in Hazewinkel, Michiel, Encyclopedia of Math-ematics, Springer, ISBN 978-1-55608-010-4

• Chatters, A. W.; Hajarnavis, C. R. (1980), Rings with chain conditions, Research Notes in Mathematics 44,Boston, Mass.: Pitman (Advanced Publishing Program), pp. vii+197, ISBN 0-273-08446-1, MR 590045

• Divinsky, N. J. (1965), Rings and radicals, Mathematical Expositions No. 14, Toronto, Ont.: University ofToronto Press, MR 0197489

• Gardner, B. J.; Wiegandt, R. (2004), Radical theory of rings, Monographs and Textbooks in Pure and AppliedMathematics 261, New York: Marcel Dekker Inc., ISBN 978-0-8247-5033-6, MR 2015465

• Goodearl, K. R. (1976), Ring theory, Marcel Dekker, ISBN 978-0-8247-6354-1, MR 0429962

• Gray, Mary (1970), A radical approach to algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0265396

• Köthe, Gottfried (1930), “Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduz-ibel ist”, Mathematische Zeitschrift 32 (1): 161–186, doi:10.1007/BF01194626

• Stenström, Bo (1971), Rings and modules of quotients, Lecture Notes in Mathematics 237, Berlin, New York:Springer-Verlag, doi:10.1007/BFb0059904, ISBN 978-3-540-05690-4, MR 0325663, Zbl 0229.16003

• Wiegandt, Richard (1974), Radical and semisimple classes of rings, Kingston, Ont.: Queen’s University, MR0349734

Chapter 29

Radical of an ideal

For other radicals, see radical of a ring.

In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x isin the radical if some power of x is in I. A radical ideal (or semiprime ideal) is an ideal that is its own radical (thiscan be phrased as being a fixed point of an operation on ideals called 'radicalization'). The radical of a primary idealis prime.Radical ideals defined here are generalized to noncommutative rings in the Semiprime ring article.

29.1 Definition

The radical of an ideal I in a commutative ring R, denoted by Rad(I) or√I , is defined as

√I = {r ∈ R | rn ∈ I for some positive integer n}.

Intuitively, one can think of the radical of I as obtained by taking all the possible roots of elements of I. Equivalently,the radical of I is the pre-image of the ideal of nilpotent elements (called nilradical) in R/I .[1] The latter shows

√I

is an ideal itself, containing I.If the radical of I is finitely generated, then some power of

√I is contained in I.[2] In particular, If I and J are ideals

of a noetherian ring, then I and J have the same radical if and only if I contains some power of J and J contains somepower of I.If an ideal I coincides with its own radical, then I is called a radical ideal or semiprime ideal.

29.2 Examples

Consider the ring Z of integers.

1. The radical of the ideal 4Z of integer multiples of 4 is 2Z.

2. The radical of 5Z is 5Z.

3. The radical of 12Z is 6Z.

4. In general, the radical of mZ is rZ, where r is the product of all distinct prime factors of m (each prime factorof m occurs exactly once as a factor of the product r) (see radical of an integer). In fact, this generalizes to anarbitrary ideal; see the properties section.

The radical of a primary ideal is prime. If the radical of an ideal I is maximal, then I is primary.[3]

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29.3. PROPERTIES 71

If I is an ideal,√In =

√I . A prime ideal is a radical ideal. So

√P = P for any prime ideal P.

Let I, J be ideals of a ring R. If√I,√J are comaximal, then I, J are comaximal.[4]

Let M be a finitely generated module over a noetherian ring R. Then

√annR(M) =

∩p∈suppM p =

∩p∈assM p [5]

where suppM is the support of M and assM is the set of associated primes of M.

29.3 Properties

This section will continue the convention that I is an ideal of a commutative ring R:

• It is always true that Rad(Rad(I))=Rad(I). Moreover, Rad(I) is the smallest radical ideal containing I.

• Rad(I) is the intersection of all the prime ideals of R that contain I. Proof: On one hand, every prime ideal isradical, and so this intersection contains Rad(I). Suppose r is an element of R which is not in Rad(I), and letS be the set {rn|n is a nonnegative integer}. By the definition of Rad(I), S must be disjoint from I. S is alsomultiplicatively closed. Thus, by a variant of Krull’s theorem, there exists a prime ideal P that contains I and isstill disjoint from S. (see prime ideal.) Since P contains I, but not r, this shows that r is not in the intersectionof prime ideals containing I. This finishes the proof. The statement may be strengthened a bit: the radical of Iis the intersection of all prime ideals of R that are minimal among those containing I.

• Specializing the last point, the nilradical (the set of all nilpotent elements) is equal to the intersection of allprime ideals of R.

• An ideal I in a ring R is radical if and only if the quotient ring R/I is reduced.

• The radical of a homogeneous ideal is homogeneous.

29.4 Applications

The primary motivation in studying radicals is the celebrated Hilbert’s Nullstellensatz in commutative algebra. Aneasily understood version of this theorem states that for an algebraically closed field k, and for any finitely generatedpolynomial ideal J in the n indeterminates x1, x2, . . . , xn over the field k, one has

I(V(J)) = Rad(J)

where

V(J) = {x ∈ kn | f(x) = 0 for all f ∈ J}

and

I(S) = {f ∈ k[x1, x2, . . . xn] | f(x) = 0 for all x ∈ S}.

Another way of putting it: The composition I(V(−)) = Rad(−) on the set of ideals of a ring is in fact a closureoperator. From the definition of the radical, it is clear that taking the radical is an idempotent operation.

72 CHAPTER 29. RADICAL OF AN IDEAL

29.5 See also• Jacobson radical

• Nilradical of a ring

29.6 Notes[1] A direct proof can be give as follows: Let a and b be in the radical of an ideal I. Then, for some positive integers m and

n, an and bm are in I. We will show that a + b is in the radical of I. Use the binomial theorem to expand (a+b)n+m−1 (withcommutativity assumed):

(a+ b)n+m−1 =

n+m−1∑i=0

(n+m− 1

i

)aibn+m−1−i.

For each i, exactly one of the following conditions will hold:

• i ≥ n• n + m − 1 − i ≥ m.

This says that in each expression aibn+m− 1 − i, either the exponent of a will be large enough to make this power of a be inI, or the exponent of b will be large enough to make this power of b be in I. Since the product of an element in I with anelement in R is in I (as I is an ideal), this product expression will be in I, and then (a+b)n+m−1 is in I, therefore a+b is in theradical of I. To finish checking that the radical is an ideal, we take an element a in the radical, with an in I and an arbitraryelement r∈R. Then, (ra)n = rnan is in I, so ra is in the radical. Thus the radical is an ideal.

[2] Atiyah–MacDonald 1969, Proposition 7.14

[3] Atiyah–MacDonald 1969, Proposition 4.2

[4] Proof: R =√√

I +√J =

√I + J implies I + J = R .

[5] Lang 2002, Ch X, Proposition 2.10

29.7 References• M.Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1994. ISBN 0-201-40751-5

• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathe-matics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

Chapter 30

Regular ideal

In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.In operator theory, a right ideal i in a (possibly) non-unital ring A is said to be regular (or modular) if there existsan element e in A such that ex− x ∈ i for every x ∈ A . (Jacobson 1956)In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor.[1] (Larsen &McCarthy 1971,p.42) This article will use “regular element ideal” to help distinguish this type of ideal.A two-sided ideal i of a ring R can also be called a (von Neumann) regular ideal if for each element x of i thereexists a y in i such that xyx=x. (Goodearl 1991, p.2) (Kaplansky 1969, p.112)Finally, regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R/J is von Neumannregular ring.[2] This article will use “quotient von Neumann regular” to refer to this type of regular ideal.Since the adjective regular has been overloaded, this article adopts the alternative adjectivesmodular, regular elementvon Neumann regular, and quotient von Neumann regular to distinguish between concepts.

30.1 Properties and examples

30.1.1 Modular ideals

The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring tonon-unital settings.A two-sided ideal i is modular if and only if A/i is unital. In a unital ring, every ideal is modular since choosing e=1works for any right ideal. So, the notion is more interesting for non-unital rings such as Banach algebras. From thedefinition it is easy to see that an ideal containing a modular ideal is itself modular.Somewhat surprisingly, it is possible to prove that even in rings without identity, a modular right ideal is contained ina maximal right ideal.[3] However, it is possible for a ring without identity to lack modular right ideals entirely.The intersection of all maximal right ideals which are modular is the Jacobson radical.[4]

Examples

• In the non-unital ring of even integers, (6) is regular ( e = 4 ) while (4) is not.• LetM be a simple right A-module. If x is a nonzero element inM, then the annihilator of x is a regular maximalright ideal in A.

• If A is a ring without maximal right ideals, then A cannot have even a single modular right ideal.

30.1.2 Regular element ideals

Every ring with unity has at least one regular element ideal: the trivial ideal R itself. Regular element ideals ofcommutative rings are essential ideals. In a semiprime right Goldie ring, the converse holds: essential ideals are all

73

74 CHAPTER 30. REGULAR IDEAL

regular element ideals. (Lam 1999, p.342)Since the product of two regular elements (=non-zerodivisors) of a commutative ring R is again a regular element, it isapparent that the product of two regular element ideals is again a regular element ideal. Clearly any ideal containinga regular element ideal is again a regular element ideal.

Examples

• In an integral domain, every nonzero element is a regular element, and so every nonzero ideal is a regularelement ideal.

• The nilradical of a commutative ring is composed entirely of nilpotent elements, and therefore no element canbe regular. This gives an example of an ideal which is not a regular element ideal.

• In an Artinian ring, each element is either invertible or a zero divisor. Because of this, such a ring only has oneregular element ideal: just R.

30.1.3 Von Neumann regular ideals

From the definition, it is clear that R is a von Neumann regular ring if and only if R is a von Neumann regular ideal.The following statement is a relevant lemma for von Neumann regular ideals:Lemma: For a ring R and proper ideal J containing an element a, there exists and element y in J such that a=aya ifand only if there exists an element r in R such that a=ara. Proof: The “only if” direction is a tautology. For the “if”direction, we have a=ara=arara. Since a is in J, so is rar, and so by setting y=rar we have the conclusion.As a consequence of this lemma, it is apparent that every ideal of a von Neumann regular ring is a von Neumannregular ideal. Another consequence is that if J and K are two ideals of R such that J⊆K and K is a von Neumannregular ideal, then J is also a von Neumann regular ideal.If J and K are two ideals of R, then K is von Neumann regular if and only if both J is a von Neumann regular idealand K/J is a von Neumann regular ring.[5]

Every ring has at least one von Neumann regular ideal, namely {0}. Furthermore, every ring has a maximal vonNeumann regular ideal containing all other von Neumann regular ideals, and this ideal is given by

M = {x ∈ R | RxR ideal regular Neumann von a is }

Examples

• As noted above, every ideal of a von Neumann regular ring is a von Neumann regular ideal.

• It is well known that a local ring which is also a von Neumann regular ring is a division ring. Let R Be alocal ring which is not a division ring, and denote the unique maximal right ideal by J. Then R cannot be vonNeumann regular, but R/J, being a division ring, is a von Neumann regular ring. Consequently, J cannot be avon Neumann regular ideal, even though it is maximal.

• A simple domain which is not a division ring has the minimum possible number of von Neumann regular ideals:only the {0} ideal.

30.1.4 Quotient von Neumann regular ideals

If J and K are quotient von Neumann regular ideals, then so is J∩K.If J⊆K are proper ideals of R and J is quotient von Neumann regular, then so is K. This is because quotients ofR/J are all von Neumann regular rings, and an isomorphism theorem for rings establishing that R/K≅(R/J)/(J/K). Inparticular if A is any ideal in R the ideal A+J is quotient von Neumann regular if J is.

Examples

30.2. REFERENCES 75

• Every proper ideal of a von Neumann regular ring is quotient von Neumann regular.

• Any maximal ideal in a commutative ring is a quotient von Neumann regular ideal since R/M is a field. Thisis not true in general because for noncommutative rings R/M may only be a simple ring, and may not be vonNeumann regular.

• Let R be a local ring which is not a division ring, and with maximal right ideal M . Then M is a quotient vonNeumann regular ideal, since R/M is a division ring, but R is not a von Neumann regular ring.

• More generally in any semilocal ring the Jacobson radical J is quotient von Neumann regular, since R/J is asemisimple ring, hence a von Neumann regular ring.

30.2 References[1] Non-zerodivisors in commutative rings are called regular elements.

[2] Burton, D.M. (1970) ``A first course in rings and ideals. Addison-Wesley. Reading, Massachusetts .

[3] Jacobson 1956, p.6.

[4] Kaplansky 1948, Lemma 1.

[5] Goodearl 1991, p.2.

• Goodearl, K. R. (1991), von Neumann regular rings (2 ed.), Malabar, FL: Robert E. Krieger Publishing Co.Inc., pp. xviii+412, ISBN 0-89464-632-X, MR 1150975

• Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications, vol.37, 190 Hope Street, Prov., R. I.: American Mathematical Society, pp. vii+263, MR 0081264

• Kaplansky, Irving (1948), “Dual rings”, Ann. of Math. (2) 49: 689–701, doi:10.2307/1969052, ISSN 0003-486X, MR 0025452

• Irving Kaplansky (1969). Fields and Rings. The University of Chicago Press.

• Larsen, Max. D.; McCarthy, Paul J. (1971). “Multiplicative theory of ideals”. Pure and Applied Mathematics(New York: Academic Press) 43: xiv;298. MR 0414528.

• Zhevlakov, K.A. (2001), “Modular ideal”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Chapter 31

Semiprime ring

A Hasse diagram of a portion of the lattice of ideals of the integers Z. The purple and green nodes indicate semiprime ideals. Thepurple nodes are prime ideals, and the purple and blue nodes are primary ideals.

In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime idealsand prime rings. In commutative algebra, semiprime ideals are also called radical ideals.For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form nZwhere n is a square-free integer. So, 30Z is a semiprime ideal of the integers, but 12Z is not.The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings.Most definitions and assertions in this article appear in (Lam 1999) and (Lam 2001).

31.1 Definitions

For a commutative ring R, a proper ideal A is a semiprime ideal if A satisfies either of the following equivalentconditions:

• If xk is in A for some positive integer k and element x of R, then x is in A.

• If y is in R but not in A, all positive integer powers of y are not in A.

The latter condition that the complement is “closed under powers” is analogous to the fact that complements of primeideals are closed under multiplication.

76

31.2. GENERAL PROPERTIES OF SEMIPRIME IDEALS 77

As with prime ideals, this is extended to noncommutative rings “ideal-wise”. The following conditions are equivalentdefinitions for a semiprime ideal A in a ring R:

• For any ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.

• For any right ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.

• For any left ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.

• For any x in R, if xRx⊆A, then x is in A.

Here again, there is a noncommutative analogue of prime ideals as complements of m-systems. A nonempty subsetS of a ring R is called an n-system if for any s in S, there exists an r in R such that srs is in S. With this notion, anadditional equivalent point may be added to the above list:

• R\A is an n-system.

The ringR is called a semiprime ring if the zero ideal is a semiprime ideal. In the commutative case, this is equivalentto R being a reduced ring, since R has no nonzero nilpotent elements. In the noncommutative case, the ring merelyhas no nonzero nilpotent right ideals. So while a reduced ring is always semiprime, the converse is not true.[1]

31.2 General properties of semiprime ideals

To begin with, it is clear that prime ideals are semiprime, and that for commutative rings, a semiprime primary idealis prime.While the intersection of prime ideals is not usually prime, it is a semiprime ideal. Shortly it will be shown that theconverse is also true, that every semiprime ideal is the intersection of a family of prime ideals.For any ideal B in a ring R, we can form the following sets:

√B :=

∩{P ⊆ R | B ⊆ P, P a prime ideal} ⊆ {x ∈ R | xn ∈ B for some n ∈ N+}

The set√B is the definition of the radical of B and is clearly a semiprime ideal containing B, and in fact is the smallest

semiprime ideal containing B. The inclusion above is sometimes proper in the general case, but for commutative ringsit becomes an equality.With this definition, an ideal A is semiprime if and only if

√A = A . At this point, it is also apparent that every

semiprime ideal is in fact the intersection of a family of prime ideals. Moreover, this shows that the intersection ofany two semiprime ideals is again semiprime.By definition R is semiprime if and only if

√{0} = {0} , that is, the intersection of all prime ideals is zero. This

ideal√{0} is also denoted by Nil∗(R) and also called Baer’s lower nilradical or the Baer-Mccoy radical or the

prime radical of R.

31.3 Semiprime Goldie rings

Main article: Goldie ring

31.4 References[1] The full ring of two-by-two matrices over a field is semiprime with nonzero nilpotent elements.

• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

78 CHAPTER 31. SEMIPRIME RING

• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), NewYork: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439

31.5 External links• PlanetMath article on semiprime ideals

Chapter 32

Spectrum of a ring

For the concept of ring spectrum in homotopy theory, see Ring spectrum.

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is the set ofall prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it intoa locally ringed space.

32.1 Zariski topology

For any ideal I of R, define VI to be the set of prime ideals containing I. We can put a topology on Spec(R) by definingthe collection of closed sets to be

{VI : I of ideal an is R}.

This topology is called the Zariski topology.A basis for the Zariski topology can be constructed as follows. For f∈R, define Df to be the set of prime ideals of Rnot containing f. Then each Df is an open subset of Spec(R), and {Df : f ∈ R} is a basis for the Zariski topology.Spec(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closedpoints in this topology. However, Spec(R) is always a Kolmogorov space. It is also a spectral space.

32.2 Sheaves and schemes

Given the space X=Spec(R) with the Zariski topology, the structure sheaf OX is defined on the Df by setting Γ(Df,OX) = Rf, the localization of R at the multiplicative system {1,f,f2,f3,...}. It can be shown that this satisfies thenecessary axioms to be a B-Sheaf. Next, if U is the union of {Dfi}i∈I, we let Γ(U,OX) = limi∈I Rfi, and thisproduces a sheaf; see the Gluing axiom article for more detail.If R is an integral domain, with field of fractions K, then we can describe the ring Γ(U,OX) more concretely as follows.We say that an element f in K is regular at a point P in X if it can be represented as a fraction f = a/b with b notin P. Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we candescribe Γ(U,OX) as precisely the set of elements of K which are regular at every point P in U.If P is a point in Spec(R), that is, a prime ideal, then the stalk at P equals the localization of R at P, and this is a localring. Consequently, Spec(R) is a locally ringed space.Every locally ringed space isomorphic to one of this form is called an affine scheme. General schemes are obtainedby “gluing together” several affine schemes.

79

80 CHAPTER 32. SPECTRUM OF A RING

32.3 Functoriality

It is useful to use the language of category theory and observe that Spec is a functor. Every ring homomorphism f :R→ S induces a continuous map Spec(f) : Spec(S) → Spec(R) (since the preimage of any prime ideal in S is a primeideal in R). In this way, Spec can be seen as a contravariant functor from the category of commutative rings to thecategory of topological spaces. Moreover for every prime P the homomorphism f descends to homomorphisms

Of −₁₍P₎ → OP

of local rings. Thus Spec even defines a contravariant functor from the category of commutative rings to the categoryof locally ringed spaces. In fact it is the universal such functor and this can be used to define the functor Spec up tonatural isomorphism.The functor Spec yields a contravariant equivalence between the category of commutative rings and the categoryof affine schemes; each of these categories is often thought of as the opposite category of the other.

32.4 Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of Kn (where K is analgebraically closed field) that are defined as the common zeros of a set of polynomials in n variables. If A is suchan algebraic set, one considers the commutative ring R of all polynomial functions A → K. The maximal ideals ofR correspond to the points of A (because K is algebraically closed), and the prime ideals of R correspond to thesubvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two properalgebraic subsets).The spectrum of R therefore consists of the points of A together with elements for all subvarieties of A. The points ofA are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all theirpoints and subvarieties. If one only considers the points of A, i.e. the maximal ideals in R, then the Zariski topologydefined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsetsas closed sets).One can thus view the topological space Spec(R) as an “enrichment” of the topological space A (with Zariski topol-ogy): for every subvariety of A, one additional non-closed point has been introduced, and this point “keeps track”of the corresponding subvariety. One thinks of this point as the generic point for the subvariety. Furthermore, thesheaf on Spec(R) and the sheaf of polynomial functions on A are essentially identical. By studying spectra of poly-nomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometryto non-algebraically closed fields and beyond, eventually arriving at the language of schemes.

32.5 Global Spec

There is a relative version of the functor Spec called global Spec, or relative Spec, and denoted by Spec. For a schemeY, and a quasi-coherent sheaf of OY-algebras A, there is a unique scheme SpecA, and a morphism f : Spec A → Ysuch that for every open affine U ⊆ Y , there is an isomorphism induced by f: f−1(U) ∼= Spec A(U) , and suchthat for open affines U ⊆ V , the inclusion f−1(U) → f−1(V ) induces the restriction map A(V ) → A(U). Thatis, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce theinclusion maps of the spectra that make up the Spec of the sheaf.

32.6 Representation theory perspective

From the perspective of representation theory, a prime ideal I corresponds to a module R/I, and the spectrum of aring corresponds to irreducible cyclic representations of R, while more general subvarieties correspond to possiblyreducible representations that need not be cyclic. Recall that abstractly, the representation theory of a group is thestudy of modules over its group algebra.The connection to representation theory is clearer if one considers the polynomial ring R = K[x1, . . . , xn] or,without a basis, R = K[V ]. As the latter formulation makes clear, a polynomial ring is the group algebra over a

32.7. FUNCTIONAL ANALYSIS PERSPECTIVE 81

vector space, and writing in terms of xi corresponds to choosing a basis for the vector space. Then an ideal I, orequivalently a module R/I, is a cyclic representation of R (cyclic meaning generated by 1 element as an R-module;this generalizes 1-dimensional representations).In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a pointin n-space, by the nullstellensatz (the maximal ideal generated by (x1 − a1), (x2 − a2), . . . , (xn − an) correspondsto the point (a1, . . . , an) ). These representations ofK[V ] are then parametrized by the dual space V ∗, the covectorbeing given by sending each xi to the corresponding ai . Thus a representation ofKn (K-linear mapsKn → K ) isgiven by a set of n numbers, or equivalently a covectorKn → K.

Thus, points in n-space, thought of as the max spec of R = K[x1, . . . , xn], correspond precisely to 1-dimensionalrepresentations of R, while finite sets of points correspond to finite-dimensional representations (which are reducible,corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal idealsthen correspond to infinite-dimensional representations.

32.7 Functional analysis perspective

Main article: Spectrum (functional analysis)For more details on this topic, see Algebra representation § Weights.

The term “spectrum” comes from the use in operator theory. Given a linear operator T on a finite-dimensional vectorspaceV, one can consider the vector space with operator as a module over the polynomial ring in one variableR=K[T],as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T](as a ring) equals the spectrum of T (as an operator).Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module)captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. Forinstance, for the 2×2 identity matrix has corresponding module:

K[T ]/(T − 1)⊕K[T ]/(T − 1)

the 2×2 zero matrix has module

K[T ]/(T − 0)⊕K[T ]/(T − 0),

showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module

K[T ]/T 2,

showing algebraic multiplicity 2 but geometric multiplicity 1.In more detail:

• the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety,with multiplicity;

• the primary decomposition of the module corresponds to the unreduced points of the variety;

• a diagonalizable (semisimple) operator corresponds to a reduced variety;

• a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit underT spans the space);

• the last invariant factor of the module equals the minimal polynomial of the operator, and the product of theinvariant factors equals the characteristic polynomial.

82 CHAPTER 32. SPECTRUM OF A RING

32.8 Generalizations

The spectrum can be generalized from rings to C*-algebras in operator theory, yielding the notion of the spectrum ofa C*-algebra. Notably, for a Hausdorff space, the algebra of scalars (the bounded continuous functions on the space,being analogous to regular functions) are a commutative C*-algebra, with the space being recovered as a topologicalspace from MSpec of the algebra of scalars, indeed functorially so; this is the content of the Banach–Stone theorem.Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yieldingthe same correspondence as between a ring and its spectrum. Generalizing to non-commutative C*-algebras yieldsnoncommutative topology.

32.9 See also• scheme

• projective scheme

• Spectrum of a matrix

• Constructible topology

• Serre’s theorem on affineness

32.10 References• Cox, David; O'Shea, Donal; Little, John (1997), Ideals, Varieties, and Algorithms, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94680-1

• Eisenbud, David; Harris, Joe (2000), The geometry of schemes, Graduate Texts in Mathematics 197, Berlin,New York: Springer-Verlag, ISBN 978-0-387-98637-1, MR 1730819

• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

32.11 External links• Kevin R. Coombes: The Spectrum of a Ring

• Miles Reid, Undergraduate Commutative Algebra, page 22

Chapter 33

Tight closure

In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positivecharacteristic. It was introduced by Melvin Hochster and Craig Huneke (1988, 1990).Let R be a commutative noetherian ring containing a field of characteristic p > 0 . Hence p is a prime number.Let I be an ideal of R . The tight closure of I , denoted by I∗ , is another ideal of R containing I . The ideal I∗ isdefined as follows.

z ∈ I∗ if and only if there exists a c ∈ R , where c is not contained in any minimal prime ideal of R ,such that czpe ∈ I [p

e] for all e ≫ 0 . If R is reduced, then one can instead consider all e > 0 .

Here I [pe] is used to denote the ideal of R generated by the pe 'th powers of elements of I , called the e th Frobeniuspower of I .An ideal is called tightly closed if I = I∗ . A ring in which all ideals are tightly closed is called weakly F -regular(for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closurecommutes with localization, and so there is the additional notion of F -regular, which says that all ideals of the ringare still tightly closed in localizations of the ring.Brenner & Monsky (2010) found a counterexample to the localization property of tight closure. However, there isstill an open question of whether every weakly F -regular ring is F -regular. That is, if every ideal in a ring is tightlyclosed, is it true that every ideal in every localization of that ring also tightly closed?

33.1 References• Brenner, Holger; Monsky, Paul (2010), “Tight closure does not commute with localization”, Annals of Math-ematics. Second Series 171 (1): 571–588, doi:10.4007/annals.2010.171.571, ISSN 0003-486X, MR 2630050

• Hochster, Melvin; Huneke, Craig (1988), “Tightly closed ideals”, American Mathematical Society. Bulletin.New Series 18 (1): 45–48, doi:10.1090/S0273-0979-1988-15592-9, ISSN 0002-9904, MR 919658

• Hochster, Melvin; Huneke, Craig (1990), “Tight closure, invariant theory, and the Briançon–Skoda theorem”,Journal of the American Mathematical Society 3 (1): 31–116, doi:10.2307/1990984, ISSN 0894-0347, MR1017784

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84 CHAPTER 33. TIGHT CLOSURE

33.2 Text and image sources, contributors, and licenses

33.2.1 Text• Annihilator (ring theory) Source: https://en.wikipedia.org/wiki/Annihilator_(ring_theory)?oldid=674571311 Contributors: Zundark,

Michael Hardy, Silverfish, Charles Matthews, Dysprosia, Alan Liefting, CyborgTosser, Waltpohl, Gauge, Rgdboer, 4v4l0n42, Lec-tonar, MarSch, SmackBot, RDBury, Silly rabbit, Nbarth, J. Finkelstein, Rschwieb, Cahk, Orko.cute, Konradek, Headbomb, Arcfrk,JackSchmidt, He7d3r, Jmanigold, Addbot, Quondum, Helpful Pixie Bot, Solomon7968, ChrisGualtieri, Lemnaminor, Hogarthian hog,M101200, Obiesel, Applezap4 and Anonymous: 17

• Ascending chain condition on principal ideals Source: https://en.wikipedia.org/wiki/Ascending_chain_condition_on_principal_ideals?oldid=643528800 Contributors: Michael Hardy, TakuyaMurata, Rjwilmsi, Algebraist, SmackBot, Rschwieb, Headbomb, RobHar, DavidEppstein, SchreiberBike, Marc van Leeuwen, MuffledThud, FrescoBot, Bezdomni~enwiki, K9re11 and Anonymous: 2

• Associated prime Source: https://en.wikipedia.org/wiki/Associated_prime?oldid=674576026 Contributors: Michael Hardy, TakuyaMu-rata, R.e.b., Rschwieb, Headbomb, David Eppstein, ClueBot, Bob1960evens, Ozob, Yobot, AnomieBOT, Solomon7968, K9re11 andAnonymous: 3

• Augmentation ideal Source: https://en.wikipedia.org/wiki/Augmentation_ideal?oldid=630819200 Contributors: Alodyne, TakuyaMu-rata, Charles Matthews, Giftlite, Ryan Reich, JPD, CRGreathouse, Vanish2, David Eppstein, Bte99, AnomieBOT, Citation bot, HSNieand Anonymous: 4

• Fractional ideal Source: https://en.wikipedia.org/wiki/Fractional_ideal?oldid=673754716 Contributors: Zundark, TakuyaMurata, Ci-phergoth, Charles Matthews, Gene Ward Smith, Waltpohl, Gauge, Gene Nygaard, Zhw, SmackBot, BlackFingolfin, Headbomb, RobHar,David Eppstein, DeaconJohnFairfax, Bogdanno, MystBot, Addbot, Expz, Luckas-bot, Erik9bot, Dewey process, WikitanvirBot, ZéroBot,Linus44 and Anonymous: 7

• Going up and going down Source: https://en.wikipedia.org/wiki/Going_up_and_going_down?oldid=649212822Contributors: Revolver,CharlesMatthews, David Shay, Giftlite, Discospinster, Paul August, Mandarax,Michael Slone, SmackBot, Rschwieb,Myasuda, Jakob.scholbach,MartinBot, LokiClock, MystBot, Addbot, Yobot, AnomieBOT, Kiefer.Wolfowitz, Jowa fan, Suslindisambiguator, Helpful Pixie Bot,K9re11 and Anonymous: 6

• Ideal (order theory) Source: https://en.wikipedia.org/wiki/Ideal_(order_theory)?oldid=676051862Contributors: Edward, Chinju, Aleph4,Robbot, MathMartin, Giftlite, Markus Krötzsch, Urhixidur, William Elliot, Schissel, Kuratowski’s Ghost, Oleg Alexandrov, Algebraist,Saintali, Trovatore, Tony1, Ott2, Modify, SmackBot, Mhss, Tsca.bot, Mets501, Oerjan, PureRumble, David Eppstein, JoergenB, Fer-nos, Manishearth, He7d3r, D.M. from Ukraine, Addbot, Luckas-bot, Ht686rg90, Labus, AnomieBOT, FrescoBot, BrideOfKripkenstein,Frietjes, Helpful Pixie Bot, Beaumont877, SindHind and Anonymous: 13

• Ideal (ring theory) Source: https://en.wikipedia.org/wiki/Ideal_(ring_theory)?oldid=651321546 Contributors: AxelBoldt, Bryan Derk-sen, Zundark, Taral, Gianfranco, Toby Bartels, Youandme, Ram-Man, Patrick, Michael Hardy, Booyabazooka, TakuyaMurata, StevanWhite, Snoyes, Ideyal, Charles Matthews, Dysprosia, Jitse Niesen, Rik Bos, Jni, Aleph4, Robbot, Mattblack82, MathMartin, TobiasBergemann, Tosha, Giftlite, Gene Ward Smith, Markus Krötzsch, Lupin, Dratman, Ssd, Waltpohl, Jorge Stolfi, Profvk, Gauss, Grunt,Noisy, Nparikh, Guanabot, Paul August, Elwikipedista~enwiki, Rgdboer, Haham hanuka, Mdd, LutzL, Jérôme, Arthena, Keenan Pep-per, Oleg Alexandrov, Graham87, Staecker, Chobot, YurikBot, Dmharvey, Grubber, Archelon, Pmdboi, -oo-~enwiki, Ms2ger, ArthurRubin, Netrapt, Pred, Prodego, Eskimbot, Dan Hoey, Bluebot, JCSantos, Bird of paradox, Lubos, Cícero, Zdenes, Mets501, Ravi12346,Rschwieb, Catherineyronwode, Zero sharp, CRGreathouse, CmdrObot, Mon4, Thijs!bot, RobHar, JAnDbot, GromXXVII, Kprateek88,Magioladitis, Reminiscenza, David Eppstein, Maurice Carbonaro, POP JAM, Jnlin, VolkovBot, Joeldl, GirasoleDE, Henry Delforn (old),PerryTachett, Functor salad, ClueBot, He7d3r, PixelBot, BOTarate, Beroal, Marc van Leeuwen, SilvonenBot, D.M. from Ukraine, Ad-dbot, Lightbot, Luckas-bot, KamikazeBot, ArthurBot, Xqbot, GrouchoBot, Charvest, FrescoBot, Citation bot 1, RedBot, Dreaming-InRed~enwiki, Dinamik-bot, Emakarov, EmausBot, John of Reading, WikitanvirBot, JasonSaulG, Quondum, Zeugding, Helpful PixieBot, Manoguru, Mdsalim786, Mcshantz, Mathedu and Anonymous: 87

• Ideal class group Source: https://en.wikipedia.org/wiki/Ideal_class_group?oldid=655790874Contributors: AxelBoldt, Zundark,MichaelHardy, Wshun, Alodyne, TakuyaMurata, Dyss, Charles Matthews, Timwi, Robbot, Georg Muntingh, Tobias Bergemann, Giftlite, GeneWard Smith, PoolGuy, Waltpohl, Pmanderson, Vivacissamamente, Gauge, Oleg Alexandrov, Rjwilmsi, Mathbot, Chobot, Dmharvey,Grubber, Eskimbot, Nbarth, Danpovey, CRGreathouse, Mon4, Thijs!bot, Headbomb, RobHar, Ilion2, Monkus2k, Smjwilson, Hesam7,Andreas Carter, DeaconJohnFairfax, Jsondow, Virginia-American, Addbot, Roentgenium111, Luckas-bot, Yobot, DemocraticLuntz, EricRowland, FrescoBot, BG19bot, Deltahedron, Enyokoyama, Progressiveforest and Anonymous: 17

• Ideal norm Source: https://en.wikipedia.org/wiki/Ideal_norm?oldid=675187459 Contributors: Michael Hardy, TakuyaMurata, CharlesMatthews, Pigsonthewing, SmackBot, Artakserkso, Rschwieb, GromXXVII, DavidCBryant, Omerks, JackSchmidt, Hatsoff, Addbot,Frobitz, Raulshc, Wcherowi, Solomon7968 and Anonymous: 6

• Ideal quotient Source: https://en.wikipedia.org/wiki/Ideal_quotient?oldid=675187295 Contributors: Charles Matthews, Stolee, Gre-gorB, Foxjwill, BlackFingolfin, Rschwieb, CBM, Expz, Yobot, Erik9bot, Uni.Liu, Solomon7968, Afshing, Citizentoad, T.Verron andAnonymous: 7

• Ideal theory Source: https://en.wikipedia.org/wiki/Ideal_theory?oldid=543901368 Contributors: Michael Hardy, Charles Matthews,Henrygb, SmackBot, JdH, JohnBlackburne, Swinburner, Addbot, Lightbot, Erik9bot, Brad7777, BattyBot and Anonymous: 2

• Jacobian ideal Source: https://en.wikipedia.org/wiki/Jacobian_ideal?oldid=532067597 Contributors: Salix alba, Racklever, Addbot andQetuth

• Jacobson radical Source: https://en.wikipedia.org/wiki/Jacobson_radical?oldid=674697253 Contributors: AxelBoldt, Michael Hardy,GTBacchus, CharlesMatthews, Aenar, Giftlite, CyborgTosser, Ssd,Waltpohl, CSTAR,Vivacissamamente, Gauge, Remag12@yahoo.com,OlegAlexandrov, Joth, Geraschenko, FRR~enwiki, Bluebot, JamieVicary,Makyen, Rschwieb, GiantSnowman, Keyi, Headbomb, Eleuther,Olaf, David Eppstein, JoergenB, VolkovBot, TXiKiBoT, Plclark, Phe-bot, ,לירן JackSchmidt, ClueBot, Alexey Muranov, Addbot, Ozob,Luckas-bot, Yobot, Gongshow, Point-set topologist, Darij, I Do Care, ElNuevoEinstein, Javierito92, Darylfunk, Jowa fan, G T Marks,Solomon7968, BattyBot, Remag12, AHusain314, Melonkelon and Anonymous: 24

33.2. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 85

• Krull’s principal ideal theorem Source: https://en.wikipedia.org/wiki/Krull’{}s_principal_ideal_theorem?oldid=649672322 Contrib-utors: Michael Hardy, TakuyaMurata, Silverfish, Charles Matthews, Giftlite, Vivacissamamente, Oleg Alexandrov, BeteNoir, Nbarth,CRGreathouse, Kummini, Cydebot, RobHar, Vanish2, Jakob.scholbach, Geometry guy, Addbot, Luckas-bot, Ulric1313, Anne Bauval,Donner60, Qetuth, K9re11 and Anonymous: 4

• Krull’s theorem Source: https://en.wikipedia.org/wiki/Krull’{}s_theorem?oldid=629506167 Contributors: TakuyaMurata, Schneelocke,Charles Matthews, MathMartin, Arthena, Ruud Koot, Gurch, Scineram, SMP, Cronholm144, Magioladitis, David Eppstein, JaGa,Alexbot, Addbot, Luckas-bot, AnomieBOT, FrescoBot, ArtemM. Pelenitsyn, Ebony Jackson, GeorgeTucker, Vincent Semeria, ZéroBot,Qetuth and Anonymous: 10

• Maximal ideal Source: https://en.wikipedia.org/wiki/Maximal_ideal?oldid=666574888 Contributors: Toby~enwiki, Michael Hardy,TakuyaMurata, Charles Matthews, Chentianran~enwiki, MathMartin, SimonMayer, Giftlite, Markus Krötzsch, Lupin, Ssd, Vivacis-samamente, Smimram, El C, EmilJ, Culix, HannsEwald, Chobot, Saintali, Hairy Dude, Grubber, DomenicDenicola, Reyk, SmackBot,Mipchunk, Viebel, JoshuaZ, Jim.belk, Rschwieb, Wafulz, Xantharius, Kprateek88, Sullivan.t.j, Error792, Flyer22, Marsupilamov, Ad-dbot, Fraggle81, TaBOT-zerem, KamikazeBot, Point-set topologist, Erik9bot, Kiefer.Wolfowitz, DixonDBot, Slawekb, ZéroBot, Maxi-malIdeal, Connor.sempek and Anonymous: 26

• Minimal ideal Source: https://en.wikipedia.org/wiki/Minimal_ideal?oldid=675189340 Contributors: Michael Hardy, Rschwieb, Ti-jfo098, Solomon7968, Remag12, Xenxax and Anonymous: 1

• Minimal prime ideal Source: https://en.wikipedia.org/wiki/Minimal_prime_ideal?oldid=646983360Contributors: TakuyaMurata, Jcw69,SmackBot, Arcfrk, JackSchmidt, Kiefer.Wolfowitz, Jonesey95, Trappist the monk, Neilme, Qetuth, Flat Out and Anonymous: 6

• Nil ideal Source: https://en.wikipedia.org/wiki/Nil_ideal?oldid=640430131 Contributors: Charles Matthews, Giftlite, Firsfron, Ian Bur-net~enwiki, Rschwieb, Headbomb, RobHar, David Eppstein, Mild Bill Hiccup, Addbot, Luckas-bot, Gongshow, Point-set topologist,Citation bot 1, ZéroBot, Helpful Pixie Bot and Anonymous: 1

• Nilpotent ideal Source: https://en.wikipedia.org/wiki/Nilpotent_ideal?oldid=610076075Contributors: CharlesMatthews, Vipul, Rjwilmsi,Rschwieb, RobHar, Addbot, Luckas-bot, Saber1983, Gongshow, Point-set topologist, Quondum and Yihedong

• Nilradical of a ring Source: https://en.wikipedia.org/wiki/Nilradical_of_a_ring?oldid=627020050 Contributors: AxelBoldt, TakuyaMu-rata, R.e.b., Rschwieb, Adam1729, VolkovBot, JackSchmidt, Nilradical, MystBot, Addbot, Ozob, ו ,.עוזי Trappist the monk, ZhongmouZhang and Anonymous: 2

• Primary ideal Source: https://en.wikipedia.org/wiki/Primary_ideal?oldid=657316713 Contributors: TakuyaMurata, Charles Matthews,Paisa, Rich Farmbrough, Keenan Pepper, Rjwilmsi, Amire80, R.e.b., Chobot, YurikBot, Dmharvey, Dan131m, Woscafrench, SmackBot,Jtwdog, Nbarth, RyanEberhart, Rschwieb, Krasnoludek, Jakob.scholbach, MetsBot, Plclark, Addbot, PV=nRT, Luckas-bot, AnomieBOT,TheLaeg, MaximalIdeal, Kammerer55, T.Verron and Anonymous: 10

• Prime ideal Source: https://en.wikipedia.org/wiki/Prime_ideal?oldid=659360112 Contributors: AxelBoldt, Bryan Derksen, Zundark,Toby~enwiki, TakuyaMurata, GTBacchus, Ciphergoth, Revolver, Charles Matthews, Topbanana, Owen, MathMartin, Giftlite, MarkusKrötzsch, Ssd, Waltpohl, Vivacissamamente, Saraedum, KronicDeth, RoyBoy, EmilJ, Jaroslavleff, Arunkumar, FlaBot, Mathbot, Mar-gosbot~enwiki, Itayperl, Ewx, Bunder, KnightRider~enwiki, SmackBot, Incnis Mrsi, Cazort, SMP, Aws4y, Jim.belk, Limaner, Rschwieb,CRGreathouse, Thijs!bot, Headbomb, JAnDbot, Albmont, Ixionid, Policron, Cajb, DorganBot, Davidsevilla, Yomcat, AlleborgoBot,GirasoleDE, SieBot, ToePeu.bot, Mr. Granger, Xingting, Addbot, AndersBot, PV=nRT, ו ,.עוזי Luckas-bot, AnomieBOT, Tchoř,Kiefer.Wolfowitz, MastiBot, RjwilmsiBot, Fly by Night, ZéroBot, Quondum, BG19bot, Ggg1390, Deltahedron, Remag12, Brirush, Kris-janus, Faizan, K9re11 and Anonymous: 46

• Primitive ideal Source: https://en.wikipedia.org/wiki/Primitive_ideal?oldid=546479411 Contributors: Silverfish, Charles Matthews,Topbanana, Paisa, CyborgTosser, Waltpohl, Bluemoose, Rschwieb, Vanish2, Addbot, Luckas-bot, Qetuth, Ggg1390 and Anonymous:1

• Principal ideal Source: https://en.wikipedia.org/wiki/Principal_ideal?oldid=674948459 Contributors: AxelBoldt, Zundark, Toby Bar-tels, Wshun, Alodyne, TakuyaMurata, Ciphergoth, Charles Matthews, Lzur, Giftlite, Ssd, Pmanderson, Vivacissamamente, EmilJ, OlegAlexandrov, Marudubshinki, Jshadias, FlaBot, Michael Slone, Reedy, SMP, Lambiam, Valoem, AndrewHowse, Mon4, WinBot, Vanish2,Albmont, Sullivan.t.j, WATARU, Pomte, VolkovBot, SieBot, Addbot, Zorrobot, DutchCanadian, ZéroBot, Quondum, Helpful Pixie Bot,Enyokoyama, Rkaup, Elysion, Bojo1498, K9re11 and Anonymous: 8

• Principal ideal theorem Source: https://en.wikipedia.org/wiki/Principal_ideal_theorem?oldid=640863107Contributors: CharlesMatthews,Giftlite, R.e.b., SmackBot, Bluebot, Markjoseph125, RobHar, VolkovBot, Geometry guy, Addbot, Brad7777, Deltahedron, Spectral se-quence, Brirush, K9re11 and Anonymous: 1

• Radical of a ring Source: https://en.wikipedia.org/wiki/Radical_of_a_ring?oldid=635819689 Contributors: Zundark, Michael Hardy,Charles Matthews, Tobias Bergemann, Giftlite, Lethe, R.e.b., Nihiltres, Algebraist, Rschwieb, Myasuda, JackSchmidt, Alexbot, Addbot,Yobot, Point-set topologist, KlioKlein, Citation bot 1, Deltahedron, Gangs of Wasseypur and Anonymous: 1

• Radical of an ideal Source: https://en.wikipedia.org/wiki/Radical_of_an_ideal?oldid=673154780 Contributors: AxelBoldt, MichaelHardy, TakuyaMurata, CharlesMatthews,Wik, Tobias Bergemann, Giftlite, Mousomer, Peruvianllama, Fpahl, GamerRemag12@aol.com,Zaslav, Gauge, Remag12@yahoo.com, Keenan Pepper, Oleg Alexandrov, Linas, Joe Decker, R.e.b., Mathbot, Chobot, Light current,Deville, KnightRider~enwiki, SmackBot, Foxjwill, Lhf, Rschwieb, Mcclurmc, Father Chaos, Thijs!bot, RobHar, SieBot, JackSchmidt,DumZiBoT, Addbot, OlEnglish, Yobot, Ptbotgourou, IhorLviv, ZéroBot, Quondum, Ferryck, PatrickR2, Teddyktchan, Saigonogetsug-atenshou and Anonymous: 22

• Regular ideal Source: https://en.wikipedia.org/wiki/Regular_ideal?oldid=621058276 Contributors: Zundark, TakuyaMurata, Rjwilmsi,Bgwhite, SmackBot, Makyen, Rschwieb, Lambtron, Addbot, Yobot, AnomieBOT, JimVC3, GrouchoBot, BG19bot, Qetuth and Markviking

• Semiprime ring Source: https://en.wikipedia.org/wiki/Semiprime_ring?oldid=674577779Contributors: TakuyaMurata, CharlesMatthews,SmackBot, Limaner, Rschwieb, Addbot, Luckas-bot, Dijkschneier, HRoestBot, Braincricket, Solomon7968 and Remag12

• Spectrum of a ring Source: https://en.wikipedia.org/wiki/Spectrum_of_a_ring?oldid=675369887 Contributors: AxelBoldt, Bryan Derk-sen, Dominus, TakuyaMurata, Schneelocke, Charles Matthews, Dysprosia, Jitse Niesen, Rvollmert, Giftlite, Fropuff, Rich Farmbrough,SamDerbyshire, Gauge, Sl, Saga City, Cuchullain, MarSch, John Z,Masnevets, Chobot, Kinser, David Pierce, Crasshopper, Marc Harper,KnightRider~enwiki, SmackBot, InverseHypercube, Nbarth, LkNsngth, Acepectif, CRGreathouse, Headbomb, Lewallen, Jakob.scholbach,

86 CHAPTER 33. TIGHT CLOSURE

JoergenB, Dr Caligari, LokiClock, SieBot, Hugh16, Lord Bruyanovich Bruyanov, Addbot, Luckas-bot, Ptbotgourou, AnomieBOT,Blaavand, SassoBot, FrescoBot, BenzolBot, Kiefer.Wolfowitz, Jonesey95, Vectornaut, Miracle Pen, Jowa fan, EmausBot, Rgugliel,Enyokoyama, Mark viking, Danneks and Anonymous: 22

• Tight closure Source: https://en.wikipedia.org/wiki/Tight_closure?oldid=627090789 Contributors: TakuyaMurata, Charles Matthews,R.e.b., Valerarren, Kummini, Arturj, TubularWorld, Sun Creator, Trappist the monk, Qetuth and Anonymous: 9

33.2.2 Images• File:A_portion_of_the_lattice_of_ideals_of_Z_illustrating_prime,_semiprime_and_primary_ideals_SVG.svg Source: https://upload.

wikimedia.org/wikipedia/commons/2/2a/A_portion_of_the_lattice_of_ideals_of_Z_illustrating_prime%2C_semiprime_and_primary_ideals_SVG.svg License: CC BY-SA 4.0 Contributors: Own work Original artist: Limaner

• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)

• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: TheTango! Desktop Project. Original artist:The people from the Tango! project. And according to themeta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

• File:Rubik’{}s_cube_v3.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b6/Rubik%27s_cube_v3.svg License: CC-BY-SA-3.0 Contributors: Image:Rubik’{}s cube v2.svg Original artist: User:Booyabazooka, User:Meph666 modified by User:Niabot

• File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)

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• Wiki_letter_w.svg Original artist: Wiki_letter_w.svg: Jarkko Piiroinen

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