Algebra and Applications

Fanggui WangHwankoo Kim

Foundations of Commutative Rings and Their Modules

Foundations of Commutative Ringsand Their Modules

Algebra and Applications

Volume 22

Series editors:

Michel Broué,Université Paris Diderot, Paris, France

Alice Fialowski,Eötvös Loránd University, Budapest, Hungary

Eric Friedlander,University of Southern California, Los Angeles, USA

Iain Gordon,University of Edinburgh, Edinburgh, UK

John Greenless,Sheffield University, Sheffield, UK

Gerhard Hiß,Aachen University, Aachen, Germany

Ieke Moerdijk,Radboud University Nijmegen, Nijmegen, The Netherlands

Christoph Schweigert,Hamburg University, Hamburg, Germany

Mina Teicher,Bar-llan University, Ramat-Gan, Israel

Alain Verschoren,University of Antwerp, Antwerp, Belgium

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http://www.springer.com/series/6253

Fanggui Wang • Hwankoo Kim

Foundations of CommutativeRings and Their Modules

123

Fanggui WangSchool of MathematicsSichuan Normal UniversityChengduChina

Hwankoo KimSchool of Computer and InformationEngineering

Hoseo UniversityAsanKorea (Republic of)

ISSN 1572-5553 ISSN 2192-2950 (electronic)Algebra and ApplicationsISBN 978-981-10-3336-0 ISBN 978-981-10-3337-7 (eBook)DOI 10.1007/978-981-10-3337-7

Library of Congress Control Number: 2016960184

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Preface

There are different approaches to characterizing the structures of commutativerings: the category of modules over commutative rings, homology theories, theoriesof the star operation on integral domains, the general theory of commutative rings,and relative homology theories. These have different emphases; however, they havesome common ground: the basic theory of commutative rings. This book starts byelaborating this theory. Thus, this book is intended to serve as a textbook for acourse in commutative algebra at a graduate level and as a reference book forresearchers.

A glance at the contents of the first five chapters shows that we cover those topicsnormally included in any commutative algebra text, although to a greater level ofdetail than other books. However, the contents in this book’s differ significantly fromthe most commutative algebra texts, namely our treatment of the Dedekind–Mertensformula (Sect. 1.7), the (small) finitistic dimension of a ring (Sect. 3.10), Gorensteinrings (Sect. 4.6), valuation overrings and the valuative dimension (Sect. 5.4), andNagata rings as quotient rings of polynomial rings (Sect. 5.5). On the other hand,Chap. 6 presents w-modules over commutative rings as they can be most commonlyused by torsion theory and multiplicative ideal theory since the w-operation theory isa bridge closely connecting torsion theory with multiplicative ideal theory. Chapter 7deals with multiplicative ideal theory over integral domains, which can be thought ofgeneralizations and extensions of work done by R. Gilmer in the bookMultiplicativeIdeal Theory [68]. In Chap. 8, we collect various results of the pullbacks, morespecially Milnor squares and D þ M constructions, which are probably the mostimportant example generating machine. In Chap. 9, we probe coherent rings withfinite weak global dimension and try to elaborate on the local ring of weak globaldimension two by combining homological tricks and methods of star operationtheory introduced in Chap. 7. Chapter 10 is devoted to the Grothendieck group of acommutative ring. In particular, the Bass–Quillen problem is discussed. Finally,Chap. 11 aims to introduce relative homological algebra, especially where therelated concepts of integral domains which appear in classical ideal theory aredefined and investigated by using the class of Gorenstein projective modules.In order to keep this book from becoming too unwieldy, we omitted important topics

vii

such as generalizations of class groups and Kronecker function rings and somegeneralizations of Krull domains.

While at least portions of the first five chapters should be read in order, theremaining chapters are essentially independent of each other, except for Chaps. 6–8.Those sections that are essentially applications of previous concepts or else are notnecessary for the rest of the book.

Each section of this book is followed by a selection of exercises, of varyingdegrees of difficulty. The exercises should deepen the reader’s understanding of theconcepts presented in this book, although some may be limited to the length of thesupplement related to content.

Chengdu, China Fanggui WangAsan, Korea (Republic of) Hwankoo KimMay 2016

viii Preface

Contents

1 Basic Theory of Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Basic Concepts of Rings and Modules . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Rings and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Basic Concepts of Modules . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Direct Product of Rings, Direct Product and Direct

Sum of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Ring Homomorphisms and Module Homomorphisms. . . . . . . . . 7

1.2.1 Ring Homomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Module Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Finitely Generated Modules and Matrix Methods . . . . . . . . . . . . 141.3.1 Finitely Generated Modules. . . . . . . . . . . . . . . . . . . . . . 141.3.2 Simple Modules, Maximal Submodules,

and Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.3 Jacobson Radical of a Ring . . . . . . . . . . . . . . . . . . . . . . 181.3.4 Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Prime Ideals and Nil Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.4.1 Prime Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.4.2 Nil Radical and Radical of an Ideal. . . . . . . . . . . . . . . . 24

1.5 Quotient Rings and Quotient Modules . . . . . . . . . . . . . . . . . . . . 251.5.1 Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.5.2 Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.5.3 Quotient Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.6 Free Modules, Torsion Modules, and Torsion-Free Modules . . . 331.6.1 Free Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.6.2 Torsion Modules, Torsion-Free Modules,

and Divisible Modules. . . . . . . . . . . . . . . . . . . . . . . . . . 361.7 Polynomial Rings and Power Series Rings . . . . . . . . . . . . . . . . . 37

1.7.1 Polynomial Rings over One Indeterminate . . . . . . . . . . 371.7.2 Polynomials with Coefficients in a Module . . . . . . . . . . 41

ix

1.7.3 Dedekind–Mertens Formula. . . . . . . . . . . . . . . . . . . . . . 421.7.4 Polynomial Rings over Many Indeterminates

and Formal Power Series Ringsover One Indeterminate . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.8 Krull Dimension of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.8.1 Basic Properties of Krull Dimension of a Ring . . . . . . . 471.8.2 Krull Dimension of a Polynomial Ring . . . . . . . . . . . . . 481.8.3 Connected Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.9 Exact Sequences and Commutative Diagrams. . . . . . . . . . . . . . . 511.9.1 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.9.2 Five Lemma and Snake Lemma . . . . . . . . . . . . . . . . . . 531.9.3 Completion of Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 591.9.4 Pushout and Pullback . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2 The Category of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.1 The Functor Hom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.1.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.1.2 Functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.1.3 Basic Properties of the Functor Hom. . . . . . . . . . . . . . . 752.1.4 Natural Transforms of Functors . . . . . . . . . . . . . . . . . . . 772.1.5 Torsionless Modules and Reflexive Modules. . . . . . . . . 78

2.2 The Functor � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.2.1 Bilinear Mappings and Tensor Products . . . . . . . . . . . . 802.2.2 Basic Properties of the Functor � . . . . . . . . . . . . . . . . . 812.2.3 Change of Rings and Adjoint

Isomorphism Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 832.2.4 Tensor Product and Localization . . . . . . . . . . . . . . . . . . 85

2.3 Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.3.1 Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.3.2 Kaplansky Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.4 Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.4.1 Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.4.2 Injective Envelope of a Module. . . . . . . . . . . . . . . . . . . 97

2.5 Flat Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.5.1 Flat Modules and Their Characterizations . . . . . . . . . . . 1002.5.2 Faithfully Flat Modules . . . . . . . . . . . . . . . . . . . . . . . . . 1062.5.3 Direct Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

2.6 Finitely Presented Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142.6.1 Finitely Presented Modules . . . . . . . . . . . . . . . . . . . . . . 1142.6.2 Isomorphism Theorems Related to Hom and �. . . . . . . 119

x Contents

2.7 Superfluous Submodules and Projective Covers . . . . . . . . . . . . . 1292.7.1 Jacobson Radical of a Module and Superfluous

Submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292.7.2 Projective Cover of a Module . . . . . . . . . . . . . . . . . . . . 130

2.8 Noetherian Modules and Artinian Modules. . . . . . . . . . . . . . . . . 1332.8.1 Noetherian Modules and Noetherian Rings . . . . . . . . . . 1332.8.2 Artinian Modules and Artinian Rings . . . . . . . . . . . . . . 135

2.9 Semisimple Modules and Composition Series. . . . . . . . . . . . . . . 1372.9.1 Semisimple Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 1372.9.2 Composition Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

2.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3 Homological Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.1 Complexes and Homologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.1.1 Complexes and Complex Morphisms . . . . . . . . . . . . . . 1473.1.2 Homology Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.1.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

3.2 Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.2.1 Comparison Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . 1553.2.2 Left Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.2.3 Right Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . 162

3.3 Derived Functor Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.3.1 Properties of Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.3.2 Dimension-Shifting Method and Isomorphism

Theorems Related to Ext . . . . . . . . . . . . . . . . . . . . . . . . 1693.4 Derived Functor Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

3.4.1 Properties of Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.4.2 Isomorphism Theorems Related to Tor . . . . . . . . . . . . . 175

3.5 Projective Dimension and Injective Dimensionof a Module and Global Dimension of a Ring. . . . . . . . . . . . . . . 1773.5.1 Projective Dimension of a Module . . . . . . . . . . . . . . . . 1773.5.2 Injective Dimension of a Module . . . . . . . . . . . . . . . . . 1793.5.3 Global Dimension of a Ring

and Semisimple Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 1803.6 Flat Dimension of a Module and Weak Global Dimension

of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1823.6.1 Flat Dimension of a Module . . . . . . . . . . . . . . . . . . . . . 1823.6.2 Weak Global Dimension of a Ring . . . . . . . . . . . . . . . . 1843.6.3 von Neumann Regular Rings. . . . . . . . . . . . . . . . . . . . . 185

3.7 Coherent Rings, Semihereditary Rings,and Hereditary Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1863.7.1 Coherent Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1863.7.2 Semihereditary Rings and Prüfer Domains . . . . . . . . . . 188

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3.7.3 Valuation Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1903.7.4 Hereditary Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.8 Change of Rings Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1933.8.1 Several Dimension Inequalities . . . . . . . . . . . . . . . . . . . 1943.8.2 Rees Theorem and Homological Dimension

of a Factor Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1963.8.3 Homological Dimension of a Polynomial Ring . . . . . . . 200

3.9 Homological Methods in Coherent Rings . . . . . . . . . . . . . . . . . . 2033.10 Finitistic Dimension of a Ring and Perfect Rings . . . . . . . . . . . . 209

3.10.1 Finitistic Dimension and Small FinitisticDimension of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

3.10.2 Semiperfect Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2123.10.3 Perfect Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

3.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4 Basic Theory of Noetherian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2254.1 Artinian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

4.1.1 Semilocal Rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2254.1.2 Basic Properties of Artinian Rings . . . . . . . . . . . . . . . . 227

4.2 Associated Prime Ideals and Primary Decompositions . . . . . . . . 2294.2.1 Associated Prime Ideals. . . . . . . . . . . . . . . . . . . . . . . . . 2294.2.2 Primary Ideals and Primary Submodules . . . . . . . . . . . . 2314.2.3 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 233

4.3 Several Classical Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2364.3.1 Injective Modules over Noetherian Rings . . . . . . . . . . . 2364.3.2 Krull’s Principal Ideal Theorem. . . . . . . . . . . . . . . . . . . 2384.3.3 Hilbert Basis Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 2414.3.4 Krull–Akizuki Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 244

4.4 Systems of Parameters and Regular Sequences. . . . . . . . . . . . . . 2484.4.1 Systems of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 2484.4.2 Regular Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2504.4.3 Auslander–Buchsbaum Theorem . . . . . . . . . . . . . . . . . . 254

4.5 Regular Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2564.5.1 Definition and Properties of Regular Local Rings . . . . . 2564.5.2 Finite Free Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . 2584.5.3 Characterizations of Regular Local Rings . . . . . . . . . . . 261

4.6 Gorenstein Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2624.6.1 QF Rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2624.6.2 n-Gorenstein Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

4.7 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

xii Contents

5 Extensions of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2735.1 Integral Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

5.1.1 Integral Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2735.1.2 GCD Domains and UFDs . . . . . . . . . . . . . . . . . . . . . . . 2775.1.3 Integrally Closed Domains . . . . . . . . . . . . . . . . . . . . . . 279

5.2 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2825.2.1 Fractional Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2825.2.2 Discrete Valuation Rings . . . . . . . . . . . . . . . . . . . . . . . . 2855.2.3 Characterizations of Dedekind Domains . . . . . . . . . . . . 287

5.3 Going Up Theorem and Going Down Theorem . . . . . . . . . . . . . 2895.3.1 Going Up Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2895.3.2 Flat Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2925.3.3 Going Down Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 294

5.4 Valuation Overrings and Valuative Dimension . . . . . . . . . . . . . . 2975.4.1 Complete Integral Closure . . . . . . . . . . . . . . . . . . . . . . . 2975.4.2 Valuation Overrings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2995.4.3 Valuative Dimension of a Ring . . . . . . . . . . . . . . . . . . . 300

5.5 Quotient Rings RhXi and R(X) of Polynomial Rings . . . . . . . . . 3025.5.1 Dimension of RhXi and RðXÞ . . . . . . . . . . . . . . . . . . . . 3025.5.2 Stably Coherent Rings. . . . . . . . . . . . . . . . . . . . . . . . . . 306

5.6 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3085.7 Valuation Methods in Rings with Zero-Divisors. . . . . . . . . . . . . 312

5.7.1 Pseudo-Localization of Rings . . . . . . . . . . . . . . . . . . . . 3125.7.2 Valuation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3135.7.3 Prüfer Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

5.8 Trivial Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

6 w-Modules over Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . 3336.1 GV-Torsion-Free Modules and w-Modules . . . . . . . . . . . . . . . . . 333

6.1.1 GV-Torsion Modulesand GV-Torsion-Free Modules . . . . . . . . . . . . . . . . . . . 333

6.1.2 w-Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3376.2 w-Closure of Modules and Prime w-Ideals . . . . . . . . . . . . . . . . . 340

6.2.1 w-Closure of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 3406.2.2 Prime w-Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

6.3 w-Exact Sequences and DW-Rings . . . . . . . . . . . . . . . . . . . . . . . 3456.3.1 w-Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3456.3.2 DW-Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

6.4 Finite Type Modules and Finitely Presented Type Modules . . . . 3496.4.1 Finite Type Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 3496.4.2 Finitely Presented Type Modules . . . . . . . . . . . . . . . . . 351

Contents xiii

6.5 w-Simple Modules and w-Semisimple Modules . . . . . . . . . . . . . 3546.5.1 w-Simple Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3546.5.2 w-Semisimple Modules . . . . . . . . . . . . . . . . . . . . . . . . . 3556.5.3 w-Jacobson Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

6.6 Quotient Ring RfXg of a Polynomial Ring R½X� . . . . . . . . . . . . 3586.6.1 GV-Ideals of a Polynomial Ring . . . . . . . . . . . . . . . . . . 3586.6.2 Properties of RfXg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

6.7 w-Flat Modules and w-Projective Modules . . . . . . . . . . . . . . . . . 3666.7.1 w-Flat Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3666.7.2 w-Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 3686.7.3 Finite Type w-Projective Modules . . . . . . . . . . . . . . . . . 372

6.8 w-Noetherian Modules and w-Noetherian Rings . . . . . . . . . . . . . 3806.8.1 Some Characterizations of w-Noetherian Rings . . . . . . . 3806.8.2 Associated Prime Ideals of a GV-Torsion-Free

Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3836.8.3 Injective Modules over w-Noetherian Rings . . . . . . . . . 3866.8.4 Krull’s Principal Ideal Theorem. . . . . . . . . . . . . . . . . . . 389

6.9 w-Artinian Modules and w-Coherent Modules . . . . . . . . . . . . . . 3906.9.1 w-Artinian Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3906.9.2 w-Coherent Modules and w-Coherent Rings . . . . . . . . . 393

6.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

7 Multiplicative Ideal Theory over Integral Domains . . . . . . . . . . . . . 4037.1 Reflexive Modules and Determinants . . . . . . . . . . . . . . . . . . . . . 403

7.1.1 Reflexive Modules over Integral Domains. . . . . . . . . . . 4037.1.2 Determinants of Torsion-Free Modules

of Finite Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4067.2 Star Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

7.2.1 Basic Properties of Star Operations . . . . . . . . . . . . . . . . 4107.2.2 �-Invertible Fractional Ideals . . . . . . . . . . . . . . . . . . . . . 413

7.3 w-Operations and w-Ideals of a Polynomial Ring . . . . . . . . . . . . 4167.3.1 w-Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4167.3.2 w-Ideals of Polynomial Rings . . . . . . . . . . . . . . . . . . . . 4187.3.3 Almost Principal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . 421

7.4 Mori Domains and Strong Mori Domains. . . . . . . . . . . . . . . . . . 4257.4.1 H-Domains and TV-Domains . . . . . . . . . . . . . . . . . . . . 4257.4.2 Mori Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4277.4.3 Strong Mori Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 428

7.5 Prüfer v-Multiplication Domains . . . . . . . . . . . . . . . . . . . . . . . . . 4307.5.1 Characterizations of PvMDs . . . . . . . . . . . . . . . . . . . . . 4307.5.2 Several (Other) Cases of Generalized Coherence. . . . . . 435

7.6 Finite Type Reflexive Modules over GCD Domains. . . . . . . . . . 4377.6.1 GCD Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4377.6.2 Finite Type Reflexive Modules

over GCD Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

xiv Contents

7.7 w-Linked Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4427.7.1 w-Linked Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4427.7.2 w-Integral Dependence and w-Integral Closure . . . . . . . 445

7.8 UMT-Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4487.9 Krull Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4537.10 Transforms of Multiplicative Systems of Ideals . . . . . . . . . . . . . 456

7.10.1 Fractional Ideals of an S-Transform . . . . . . . . . . . . . . . 4577.10.2 Global Transforms and w-Global Transforms . . . . . . . . 4597.10.3 Mori–Nagata Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 461

7.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

8 Structural Theory of Milnor Squares . . . . . . . . . . . . . . . . . . . . . . . . 4698.1 Basic Properties of Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

8.1.1 Pullbacks of Rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4698.1.2 Pullbacks of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 471

8.2 Homological Properties of Cartesian Squares . . . . . . . . . . . . . . . 4758.2.1 Pullbacks of Flat Modules . . . . . . . . . . . . . . . . . . . . . . . 4758.2.2 Pullbacks of Projective Modules . . . . . . . . . . . . . . . . . . 4788.2.3 Finiteness Conditions and Coherence in Cartesian

Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4808.3 Basic Properties of Milnor Squares. . . . . . . . . . . . . . . . . . . . . . . 482

8.3.1 Localization Methods in Milnor Squares . . . . . . . . . . . . 4828.3.2 Star Operation Methods in Milnor Squares . . . . . . . . . . 4858.3.3 Prime Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4898.3.4 Weak Finiteness Conditions in Milnor Squares . . . . . . . 491

8.4 Chain Conditions of Rings in Milnor Squares . . . . . . . . . . . . . . 4928.4.1 Pullbacks of Mori Domains. . . . . . . . . . . . . . . . . . . . . . 4938.4.2 Pullbacks of Noetherian Domains

and SM Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4978.5 Coherence of Rings in Milnor Squares . . . . . . . . . . . . . . . . . . . . 500

8.5.1 Pullbacks of v-Coherent Domains . . . . . . . . . . . . . . . . . 5008.5.2 Pullbacks of Coherent Rings . . . . . . . . . . . . . . . . . . . . . 5058.5.3 Pullbacks of Quasi-Coherent Domains

and FC Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5088.5.4 Pullbacks of w-Coherent Domains, w-Quasi-Coherent

Domains, and WFC Domains . . . . . . . . . . . . . . . . . . . . 5108.6 Integrality and w-Invertibility in Milnor Squares . . . . . . . . . . . . 512

8.6.1 Pullbacks of Prüfer Domains and PvMDs . . . . . . . . . . . 5128.6.2 Integrally Closedness in Milnor Squares . . . . . . . . . . . . 5148.6.3 Pullbacks of UMT-Domains . . . . . . . . . . . . . . . . . . . . . 5158.6.4 Basic Properties of DþM Constructions. . . . . . . . . . . . 5178.6.5 Pullbacks of GCD Domains . . . . . . . . . . . . . . . . . . . . . 518

Contents xv

8.7 Dimensions of Rings in Milnor Squares . . . . . . . . . . . . . . . . . . . 5208.7.1 Krull Dimensions of Rings in Milnor Squares. . . . . . . . 5208.7.2 w-Dimensions of Rings in Milnor Squares . . . . . . . . . . 5228.7.3 Valuative Dimensions in Milnor Squares. . . . . . . . . . . . 5248.7.4 t-Dimensions of Rings in Milnor Squares . . . . . . . . . . . 528

8.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

9 Coherent Rings with Finite Weak Global Dimension . . . . . . . . . . . . 5359.1 Fitting Invariant Ideals and Coherent Regular Rings. . . . . . . . . . 535

9.1.1 Fitting Invariant Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . 5359.1.2 w-Ideals of Coherent Regular Rings . . . . . . . . . . . . . . . 542

9.2 Super Coherent Regular Local Rings and GeneralizedUmbrella Rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5439.2.1 Super Coherent Regular Local Rings. . . . . . . . . . . . . . . 5439.2.2 Generalized Umbrella Rings . . . . . . . . . . . . . . . . . . . . . 552

9.3 Domains with Weak Global Dimension 2. . . . . . . . . . . . . . . . . . 5549.4 Umbrella Rings and U2-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

9.4.1 Structural Characterizations of Umbrella Rings . . . . . . . 5589.4.2 Properties of U2-rings . . . . . . . . . . . . . . . . . . . . . . . . . . 561

9.5 GE Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5639.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

10 The Grothendieck Group of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . 57310.1 Basic Properties of Grothendieck Groups . . . . . . . . . . . . . . . . . . 57310.2 Picard Groups of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

10.2.1 Invertible Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57910.2.2 Exterior Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

10.3 Grothendieck Groups of Dedekind Domains. . . . . . . . . . . . . . . . 58810.4 Grothendieck Groups of Polynomial Rings. . . . . . . . . . . . . . . . . 598

10.4.1 Grothendieck Groups in the Category of FinitelyPresented Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

10.4.2 Grothendieck Groups of Polynomial Rings . . . . . . . . . . 59910.5 The Bass–Quillen Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

10.5.1 Gluing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60310.5.2 Bass–Quillen Conjecture and Quillen’s Method . . . . . . 60810.5.3 Lequain–Simis Method . . . . . . . . . . . . . . . . . . . . . . . . . 613

10.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

11 Relative Homological Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61911.1 Gorenstein Projective Modules and Strongly Gorenstein

Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61911.1.1 Gorenstein Projective Modules . . . . . . . . . . . . . . . . . . . 61911.1.2 Strongly Gorenstein Projective Modules . . . . . . . . . . . . 62211.1.3 n-Strongly Gorenstein Projective Modules. . . . . . . . . . . 628

xvi Contents

11.2 Gorenstein Injective Modules and Strongly GorensteinInjective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63211.2.1 Gorenstein Injective Modules . . . . . . . . . . . . . . . . . . . . 63211.2.2 Strongly Gorenstein Injective Modules . . . . . . . . . . . . . 63311.2.3 n-Strongly Gorenstein Injective Modules. . . . . . . . . . . . 634

11.3 Gorenstein Projective Dimension and Gorenstein InjectiveDimension of a Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63511.3.1 Gorenstein Projective Dimension of a Module . . . . . . . 63511.3.2 Gorenstein Injective Dimension of a Module . . . . . . . . 642

11.4 Gorenstein Global Dimension of a Ring . . . . . . . . . . . . . . . . . . . 64411.4.1 Basic Properties of the Gorenstein Global Dimension

of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64411.4.2 Rings of Gorenstein Global Dimension 0 . . . . . . . . . . . 647

11.5 Change of Rings Theorems for the Gorenstein ProjectiveDimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64911.5.1 Gorenstein Global Dimension of a Factor Ring. . . . . . . 64911.5.2 Gorenstein Global Dimension

of a Polynomial Ring . . . . . . . . . . . . . . . . . . . . . . . . . . 65111.6 Finitely Generated Gorenstein Projective Modules . . . . . . . . . . . 653

11.6.1 Super Finitely Presented Modules . . . . . . . . . . . . . . . . . 65311.6.2 Finitely Generated Gorenstein Projective Modules . . . . 656

11.7 Gorenstein Hereditary Rings and Gorenstein DedekindDomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66211.7.1 Gorenstein Hereditary Rings . . . . . . . . . . . . . . . . . . . . . 66211.7.2 Gorenstein Dedekind Domains . . . . . . . . . . . . . . . . . . . 66711.7.3 Noetherian Warfield Domains . . . . . . . . . . . . . . . . . . . . 669

11.8 Pseudo Valuation Rings and 2-Discrete Valuation Rings . . . . . . 67111.8.1 Pseudo Valuation Rings . . . . . . . . . . . . . . . . . . . . . . . . 67211.8.2 2-Discrete Valuation Rings . . . . . . . . . . . . . . . . . . . . . . 677

11.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

Contents xvii

Symbols

Z The set of integers, Sect. 1.1N The set of natural numbers, Sect. 1.1Zþ The set of positive integers, Sect. 1.1Q The set of rational numbers, Sect. 1.1R The set of reals numbers, Sect. 1.1A�B A is a subset of B, Sect. 1.1Cnm The coefficient of X

n in the expansion of ð1þXÞm, Sect. 1.1f jA The restriction of a map f on a set A, Sect. 1.2kerð f Þ The kernel of a homomorphism f , Sect. 1.2Imð f Þ The image of a homomorphism f , Sect. 1.2Cokerð f Þ The cokernel of a homomorphism f , Sect. 1.2; The empty set, Sect. 1.3jAj The cardinality of a set A, Sect. 1.3JðRÞ The Jacobson radical of a ring R, Sect. 1.3nilðRÞ The nil radical of a ring R, Sect. 1.3MaxðRÞ The set of maximal ideals of a ring R, Sect. 1.3detðAÞ The determinant of a matrix A, Sect. 1.3adjðAÞ The adjoint of a matrix A, Sect. 1.3x 2 AnB x 2 A, but x 62 B, Sect. 1.4x 2 Anf0g x 2 A; x 6¼ 0, Sect. 1.4A � B A is a proper subset of B, Sect. 1.4TðRÞ The total quotient ring of a ring R, Sect. 1.5qf ðRÞ The quotient field of a domain R, Sect. 1.5dimKðVÞ The dimension of a vector space V over a field K, Sect. 1.5rankðAÞ The rank of a matrix A, Sect. 1.6rankðFÞ The rank of a free module F, Sect. 1.6cð f Þ The content of a polynomial f , Sect. 1.7dimðRÞ The Krull dimension of a ring R, Sect. 1.8dimvðRÞ The valuative dimension of a domain R, Sect. 1.8SpecðRÞ The set of prime ideals of a ring R, Sect. 1.8

xix

M The category of modules, Sect. 2.1M� (¼ HomRðM;RÞ), the dual of a module M, Sect. 2.1M�� (¼ HomRðHomRðM;RÞ;RÞ), the double dual of a module M,

Sect. 2.1pdRM The projective dimension of an R-module M, Sect. 3.5idRN The injective dimension of an R-module N, Sect. 3.5gl:dimðRÞ The global dimension of a ring R, Sect. 3.5fdRM The flat dimension of an R-module M, Sect. 3.6w:gl:dimðRÞ The weak global dimension of a ring R, Sect. 3.5M þ (¼ HomðM;Q=ZÞ), the characteristic module of a module M,

Sect. 3.11rankðMÞ The rank of a module M over a domain, Sect. 4.3w-MaxðRÞ The set of maximal w-ideals of a ring R, Sect. 6.2Q0ðRÞ The ring of finite fractions of a ring R, Sect. 6.6detðMÞ The determinant of a module M, Sect. 7.1G-pdRM The Gorenstein projective dimension of an R-module M, Sect. 11.3G-idRN The Gorenstein injective dimension of an R-module N, Sect. 11.3G-gl:dimðRÞ The Gorenstein global dimension of a ring R, Sect. 11.4

xx Symbols

Chapter 1Basic Theory of Rings and Modules

All the rings in this book are commutative. Sometimes we may also assume that analgebra may be non-commutative. In this chapter we introduce the basic terminologyfor rings and modules. For later use, we present some preliminary definitions andresults: the fundamental theorems of homomorphisms of rings and modules; localrings and localization methods; the height of a prime ideal and the Krull dimensionof a ring; fractional ideals over a domain. We also include Nakayama’s lemma andthe Dedekind–Mertens formula.

Modern mathematics is inseparable from the concept of modules. Modules canbe considered a natural extension of Abelian groups and vector spaces. Since the the-ory of modules is the basis for algebraic theory, we need to study it systematically.This chapter introduces some basic concepts of modules, including modules andsubmodules, module homomorphisms, direct products and direct sums, finitely gen-erated modules, etc. To save space and to promote greater insight, we blend moduletheory methods and ideal theory methods together. Since the ideas of classificationof modules and the theory of vector spaces are linked, understanding of free modulesis most basic. In order to understand the category of modules better, we also intro-duce short exact sequences and the five lemma. By studying modules, the reader willencounter module homomorphisms, exact sequences, and the important role of thecommutative diagram as demonstrated in the problem discussion.

1.1 Basic Concepts of Rings and Modules

1.1.1 Rings and Ideals

Let R be a ring. In this book we denote by 1 the identity element of R. We denote byN, Z, Z+, Q, and R the set of natural numbers, the set of integers, the set of positive

© Springer Nature Singapore Pte Ltd. 2016F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules,Algebra and Applications 22, DOI 10.1007/978-981-10-3337-7_1

1

2 1 Basic Theory of Rings and Modules

integers, the set of rational numbers, and the set of real numbers respectively. Thering Z of integers and the polynomial ring F[X1, . . . , Xn] over a number field F areexamples of rings. We are allowed to have 1 = 0 in a ring, but if this happens, thenR has only one element 0. In the statements of definitions and theorems about rings,we often omit the condition 1 �= 0. An element a ∈ R is called a unit (or invertibleelement) of R if there exists an element b ∈ R such that ab = 1; in this case, b iscertainly unique and we write b = a−1, which is called the inverse of a. It is obviousthat the set of units of R is a group under multiplication, which is called the unitgroup of R and is denoted by U (R).

We say that an element a ∈ R is a zero-divisor if there is a nonzero elementb ∈ R such that ab = 0. If a ∈ R is not a zero-divisor, then we say that a is anon-zero-divisor or a regular element. We call an element a ∈ R nilpotent if thereis a positive integer n such that an = 0. We say that R is a domain if R has nononzero zero-divisors. If all nonzero elements of R are units, then we say that R isa field. Naturally, fields are domains, but a domain is not necessarily a field. Whenwe state that R is a domain, it may happen that the condition that R is not a field isomitted, depending on the context, even when it is actually necessary.

For nonnegative integers n and k, the binomial coefficient Ckn can be defined asthe coefficient of the monomial Xk in the expansion of (1 + X)n .Example 1.1.1 (1) All rings satisfy the binomial theorem, that is, for any a, b ∈ R

(a + b)n =n∑

i=0Cina

n−i bi .

(2) If a, b ∈ R are nilpotent elements of R, then a + b is also nilpotent. In fact,let an = 0 and bm = 0 for some positive integers m and n, then

(a + b)n+m =m∑

i=0Cin+ma

n+m−i bi +n+m∑

i=m+1Cin+ma

n+m−i bi = 0.

In general, if a1, a2, . . . , an are nilpotent elements, then a1+a2+· · ·+an is nilpotent.(3) If u, a ∈ R, u is a unit, and a is a nilpotent element, then it is easy to see that

u + a is a unit.Let I be a nonempty subset of R. We say that I is an ideal of R if a + b ∈ I and

ra ∈ I for all a, b ∈ I and all r ∈ R (notice that R has the identity element). It isclear that I := {0} (also write I = 0 and call it the zero ideal) and I := R are idealsof R. These two ideals are said to be trivial. If I and J are ideals of R with I ⊆ J ,then we say that I is a subideal of J .

Example 1.1.2 (1) Let I be an ideal of R. Then I = R if and only if I contains aunit of R.

(2) A ring R is a field if and only if R has only the trivial ideals.

1.1 Basic Concepts of Rings and Modules 3

Let I be an ideal of R. For a ∈ R, denote by a the residue class (or coset) of a,that is, a = a + I = {a + u | u ∈ I }. The set R/I of residue classes becomes also aring under the addition and the multiplication

a + b = a + b and a b = ab,

which is called the factor ring of R modulo the ideal I (or by I ).Let T be a ring and let R be a subset of T . Then R is called a subring of T if 1 ∈ R

and the addition and multiplication of T make R become a ring. Correspondingly,T is called an extension of R. Since we require a subring to have identity 1, an idealI is a subring of R if and only if I = R; a subring R is an ideal of T if and only ifR = T .

1.1.2 Basic Concepts of Modules

Definition 1.1.3 Let M be an Abelian group. Then M is called an R-module if thereexists a map R × M → M , by (a, x) �→ ax , satisfying the following conditions:For any a, b ∈ R and x, y ∈ M ,

(1) (ab)x = a(bx).(2) (a + b)x = ax + bx .(3) a(x + y) = ax + ay.(4) 1x = x .

Example 1.1.4 (1) If R is a field, then the concept of R-modules is the same as thatof R-vector spaces.

(2) If R = Z, then the concept of R-modules is the same as that of Abelian groups.Example 1.1.5 A ring R itself is an R-module. More generally, every ideal of R isan R-module.

Example 1.1.6 Since everymodule has a zero element, ifM is themodule constitutedby only one element, then the element must be the zero element. In this case, we callM the zero module and write M = {0} or M = 0.Example 1.1.7 Let M be an R-module. Then for a, ai ∈ R and x, xi ∈ M ,

(1) a0 = 0 and 0x = 0.(2) a(−x) = −ax and (−a)x = −ax .(3) a(

∑xi ) = ∑ axi .

(4) (∑

ai )x = ∑ ai x .Definition 1.1.8 Let M be an R-module and let N be a nonempty subset of M . ThenN is called a submodule of M if for any a ∈ R and x, y ∈ N , we have x + y ∈ Nand ax ∈ N , that is, N itself is an R-module.

4 1 Basic Theory of Rings and Modules

Example 1.1.9 (1) If M is a module, then 0 and M themselves are submodules ofM , called the trivial submodules of M .

(2) Every ideal of a ring R is a submodule of R, as an R-module.

Example 1.1.10 Let {Mi } be a family of submodules of a module M . Then ⋂iMi

is also a submodule of M . In particular, let {Ii | i ∈ Γ } be a family of ideals of R.Then

⋂i∈Γ

Ii is also an ideal of R.

Example 1.1.11 Let M be an R-module and let M1, M2, . . . , Mn be submodules ofM . Define

M1 + M2 + · · · + Mn = {x1 + x2 + · · · + xn | xi ∈ Mi , i = 1, 2, . . . , n}.

It is easy to see that M1 +M2 +· · ·+Mn is a submodule of M , which is called thesum ofM1, M2, . . . , Mn .More generally, let {Mi | i ∈ Γ } be a family of submodulesof M . Define

∑

i

Mi = {xi1 + xi2 + · · · + xin | n ∈ Z+, xik ∈ Mik , ik ∈ Γ }.

Then∑i∈Γ

Mi is a submodule of M , which is called the sum of {Mi | i ∈ Γ }.

Since every ideal of R is a submodule of R, the above sum of submodules can beapplied to R. Therefore we have the same definition of the sum of ideals of R. Wedo not have to repeat it.

An element x of the sum∑i∈Γ

Mi of submodules can be often expressed as x =∑i∈Γ

xi , where xi ∈ Mi and only finitely many xi �= 0.

Example 1.1.12 Let I be an ideal of R and let M be an R-module. Define

I M = {m∑

i=1ai xi | m ∈ Z+, ai ∈ I, xi ∈ M}.

Then I M is a submodule of M , which is called the product of I and M .Sometimes it is convenient to write a general element of I M as

∑i ai xi , where

ai ∈ I , xi ∈ M , and indicate that only a finite number of ai xi �= 0.If I and J are ideals of R, we also have the same argument of the product I J of

two ideals of R. But we can continue this approach. More generally, let I1, I2, . . . , Inbe ideals of R. Define inductively:

I1 I2 · · · In = (I1 I2 · · · In−1)In.

1.1 Basic Concepts of Rings and Modules 5

Naturally I1 I2 · · · In ⊆ I1 ∩ I2 ∩ · · · ∩ In . Given an ideal I of R, we define:

I 0 = R, I 1 = I, I 2 = I I, . . . , I n = (I n−1)I.

Let N be a submodule of M . We have a factor group M/N. For x ∈ M , use x todenote the coset x + N . So inM/N,

x + y = x + y, x, y ∈ M.

For r ∈ R, definer x = r x, r ∈ R, x ∈ M.

Then M/N becomes an R-module, which is called the factor module of M moduloN (or by N ).

Example 1.1.13 Let I be an ideal of R, M be an R-module, and R = R/I . Define

r x = r x, r ∈ R, x ∈ M.

Then an R-module M = M/I M is an R-module.Let M be an R-module and s ∈ R. Then s is called a zero-divisor of M if there

exists x ∈ M\{0} such that sx = 0. Naturally s = 0 is called the trivial zero-divisorof M . Let X be a subset of M . For r ∈ R, write r X = {r x | x ∈ X}. Define

annR(X) = {r ∈ R | r X = 0},

which is an ideal of R, called the annihilator of X . Correspondingly, for a subset Aof R, define

annM(A) = {x ∈ M | Ax = 0},

which is a submodule of M , called the annihilator of A. If no confusion arises, wewrite ann(X) and ann(A) for annR(X) and annM(A), respectively.

Example 1.1.14 (1) Let M be an R-module. If ann(M) = 0, then M is called afaithful module.

(2) If I is an ideal of R and M is an R-module with I M = 0, then I ⊆ ann(M).Set R = R/I and define

r x = r x, r ∈ R, x ∈ M.

Then M becomes an R-module.(3) Let M be an R-module, A be a submodule of M , and B be a nonzero subset

of M . Set(A : B) = {r ∈ R | r B ⊆ A}.

6 1 Basic Theory of Rings and Modules

Then (A : B) is an ideal of R, which is called the residual of A by B. If B = {x},we abbreviate this as (A : x).

(4) Let A be a submodule of M . Then (A : M) = ann(M/A). If x ∈ M , wedenote by x the image of x in M/A. Then (A : x) = ann(x).

1.1.3 Direct Product of Rings, Direct Product and DirectSum of Modules

Let {Ri | i ∈ Γ } be a family of rings. Consider the Cartesian product ∏iRi . Then

every element in∏iRi is understood as a vector on the index setΓ , referred to as [ri ],

where ri ∈ Ri . For each k ∈ Γ , rk is referred to as the corresponding component ofthe index k, or generally called the k-th component of the vector [ri ]. For ai , bi ∈ Ri ,define

[ai ] + [bi ] = [ai + bi ],

[ai ][bi ] = [aibi ].

In this case,∏iRi becomes a ring, which is called the direct product of a family of

rings {Ri | i ∈ Γ }. Its zero element is the element whose i-th component is the zeroof Ri for all i , while its identity element is the element whose i-th component is theidentity of Ri for all i (note: the identity element of Ri is still 1). In particular, in thedirect product R1×· · ·×Rn of rings R1, . . . , Rn , the zero element is 0 = (0, 0, . . . , 0)and the identity element is 1 = (1, 1, . . . , 1).

Let R be a ring and let {Mi | i ∈ Γ } be a family of R-modules. Define∏

i

Mi = {[xi ] | xi ∈ Mi , i ∈ Γ }.

Here, every element in∏iMi is understood as a vector on the index set Γ . In

∏iMi ,

define:[xi ] + [yi ] = [xi + yi ],

r [xi ] = [r xi ].

Then∏iMi becomes an R-module, which is called the direct product of the family

{Mi }. In the case that Mi = M for all i ∈ Γ , we write MΓ for ∏iMi .

Then define

⊕

i

Mi = {[xi ] | xi ∈ Mi , xi �= 0 only for finitely many i ∈ Γ }.

1.1 Basic Concepts of Rings and Modules 7

Then⊕iMi is a submodule of

∏iMi , which is called the direct sum of the family

{Mi }. In the case that Mi = M for all i ∈ Γ , we write M (Γ ) for ⊕iMi .

It is easy to see that if Γ is a finite set, then direct products and direct sumscoincide.

1.2 Ring Homomorphisms and Module Homomorphisms

1.2.1 Ring Homomorphisms

Definition 1.2.1 Let R and T be rings. Then a map f : R → T is called a (ring)homomorphism if f satisfies: For any a, b ∈ R,

(1) f (a + b) = f (a) + f (b),(2) f (ab) = f (a) f (b), and(3) f (1) = 1.

Example 1.2.2 Let R and T be rings and let f : R → T be a homomorphism.(1) f (0) = 0 and for any x, y ∈ R, we have f (x − y) = f (x) − f (y).(2) If u is a unit of R, then f (u) is a unit of T .

Let f : R → T be a (ring) homomorphism. Then f is called a monomorphismif f is injective; f is called an epimorphism if f is surjective; f is called anisomorphism if f is bijective. In the last case, we say that R is isomorphic to T ,denoted by R ∼= T .

Note that isomorphism between rings is an equivalence relation.

Example 1.2.3 (1) For any ring R, the identity map

1R : R → R, 1R(r) = r, r ∈ R

is an isomorphism.(2) If R is a subring of a ring T , then the inclusion map

λ : R → T, λ(r) = r, r ∈ R

is a monomorphism.(3) If I is an ideal of R, then the natural map

π : R → R/I, π(r) = r , r ∈ R

is an epimorphism. We call π the natural homomorphism from R to R/I.(4) Let R1 and R2 be rings, r1 ∈ R1, r2 ∈ R2, p1(r1, r2) = r1, p2(r1, r2) = r2.

Then pi (i = 1, 2) is an epimorphism, which is called the projection on the i-thcomponent.

8 1 Basic Theory of Rings and Modules

Let R and T be rings and let f : R → T be a homomorphism. The set

Ker( f ) := {x ∈ R | f (x) = 0}

is called the kernel of f .

Theorem 1.2.4 Let f : R → T be a ring homomorphism. Then:(1) Ker( f ) is a proper ideal of R.(2) f is a monomorphism if and only if Ker( f ) = 0.

Proof (1) Since f (1) = 1 �= 0, it follows that Ker( f ) �= R. Let a, b ∈ Ker( f ).Then f (a) = f (b) = 0. Thus f (a + b) = f (a) + f (b) = 0 + 0 = 0, andso a + b ∈ Ker( f ). If r ∈ R, then f (ra) = f (r) f (a) = f (r)0 = 0. Thusra ∈ Ker( f ). Therefore, Ker( f ) is a proper ideal of R.

(2) If f is a monomorphism and x ∈ Ker( f ), then f (x) = 0 = f (0), and sox = 0. Therefore, Ker( f ) = 0.

Conversely, suppose that Ker( f ) = 0. If f (a) = f (b) for a, b ∈ R, then f (a −b) = f (a) − f (b) = 0, and so a − b ∈ Ker( f ). Thus a − b = 0. Therefore, f is amonomorphism. �

Commutative Diagram of Maps. Let A, B,C be sets, f : A → C , g : A → B,h : B → C be maps. Then the following diagram

Af ��

g���

����

��C

B

h

���������

is said to be commutative if hg = f holds. Similarly let A, B,C, D be sets, f :A → B, g : B → D, h : A → C , k : C → D be maps. Then the following diagram

Af ��

h

��

B

g

��C

k�� D

is said to be commutative if g f = kh holds.Theorem 1.2.5 (Fundamental Theorem on Homomorphisms) Let f : R → T be ahomomorphism, I be an ideal of R, and R := R/I . If I ⊆ Ker( f ), then there existsa unique homomorphism f : R → T , making the following diagram commute:

1.2 Ring Homomorphisms and Module Homomorphisms 9

Rf ��

π �����

���� T

Rf

����

��

In particular, f is a monomorphism if and only if I = Ker( f ).Proof For any x ∈ R, set f (x) = f (x). Then f is well-defined. Indeed, if x = y,then x + I = y + I , so x − y ∈ I ⊆ Ker( f ), and thus f (x − y) = 0. Hencef (x) = f (y). Trivially, f is a homomorphism. For the uniqueness, if g : R → Tsuch that gπ = f , then for any x ∈ R, g(x) = gπ(x) = f (x) = f π(x) = f (x).Thus g = f .

Suppose that f is amonomorphism. If x ∈ Ker( f ), then f (x) = f (x) = 0.Hencex = 0. Thus x ∈ I . So I = Ker( f ). Conversely, suppose that I = Ker( f ). If f (x) =f (x) = 0, then x ∈ Ker( f ) = I , that is, x = 0. Thus f is a monomorphism. �Corollary 1.2.6 (First Isomorphism Theorem) If f : R → T is an epimorphism,then

R/Ker( f ) ∼= T .

Theorem 1.2.7 (Second Isomorphism Theorem) Let R be a ring, T be an extensionof R, and N be an ideal of T . Then R+ N is a subring of T , N is an ideal of R+ N,N ∩ R is an ideal of R, and

(R + N )/N ∼= R/(N ∩ R).

Proof The assertions of the first three statements are obvious. For any x ∈ R, definef : R → (R + N )/N by f (x) = x + N . Then it is easy to see that f is anepimorphism and Ker( f ) = N ∩ R. �Theorem 1.2.8 (Third Isomorphism Theorem) Let R be a ring and let N , H beproper ideals of R with N ⊆ H. Then

R/H ∼= (R/N )/(H/N ).

Proof Define f : R/N → R/H by f (x + N ) = x + H for any x ∈ R. SinceN ⊆ H , we have f is a homomorphism. It is easy to see that f is an epimorphismand Ker( f ) = H/N . �Definition 1.2.9 Let I and J be ideals of R. Then I and J are said to be relativelyprime or comaximal if I + J = R.Lemma 1.2.10 (1) If an ideal I and ideals J1, J2 are relatively prime, then I andJ1 J2 are relatively prime. More generally, if each ideal Ii (i = 1, . . . , n) and each

10 1 Basic Theory of Rings and Modules

ideal J j ( j = 1, . . . ,m) are relatively prime, then I1 · · · In and J1 · · · Jm are rela-tively prime.

(2) If I1, I2, . . . , In are relatively prime, then

I1 ∩ I2 ∩ · · · ∩ In = I1 I2 · · · In.

Proof (1) By the hypothesis, I + J1 = R and I + J2 = R. Then

R = R2 = (I + J1)(I + J2) = I 2 + I J1 + I J2 + J1 J2 ⊆ I + J1 J2 ⊆ R.

Therefore, I + J1 J2 = R.(2) By (1), it is enough to prove the case n = 2. Let x ∈ I1∩ I2. Since I1+ I2 = R,

there exist a ∈ I1 and b ∈ I2 such that 1 = a + b. Thus x = xa + xb ∈ I1 I2, and soI1 ∩ I2 = I1 I2. �Theorem 1.2.11 (Chinese Remainder Theorem) Let I1, I2, . . . , In be ideals of Rwhich are relatively prime. Write I = I1 ∩ I2 ∩ · · · ∩ In. Then the map

θ : R/I →n∏

i=1R/Ii , θ(r + I ) = (r + I1, r + I2, . . . , r + In)

is a ring isomorphism.

Proof Define f : R →n∏

i=1R/Ii by f (r) = (r + I1, r + I2, . . . , r + In) for r ∈ R.

Then f is a ring homomorphism. Note that r ∈ Ker( f ) if and only if r + Ii = 0for any i ; if and only if r ∈ ⋂

iIi = I . Thus I = Ker( f ). Note that f induces a

homomorphism θ : R/I →n∏

i=1R/Ii .

Let r1, r2, . . . , rn ∈ R. For each k, by Lemma 1.2.10, Ik and I1 · · · Ik−1 Ik+1 · · · Inare relatively prime.Thus1 = ak+bk ,whereak ∈ Ik andbk ∈ I1 · · · Ik−1 Ik+1 · · · In =n⋂

i �=kIi . Also rk = rkak + ck , where ck = rkbk ∈

n⋂i �=k

Ii , k = 1, 2, . . . , n. Setr = c1 + c2 + · · · + cn . Since rkak ∈ Ik and ci ∈ Ik for all i �= k, we getr + Ik = ck + Ik = rk + Ik . Therefore, θ is an epimorphism. Note that the inducedhomomorphism R/I →

n∏i=1

R/Ii of f is exactly θ . By Corollary 1.2.6, θ is an

isomorphism. �

1.2.2 Module Homomorphisms

Let R be a ring. To discuss the relationship between two R-modules, naturally weneed the concept of module homomorphisms.

1.2 Ring Homomorphisms and Module Homomorphisms 11

Definition 1.2.12 LetM andM ′ be R-modules and let f : M → M ′ be amap. Thenf is called a (module) homomorphism from M to M ′ if it satisfies the followingconditions: for any x, y ∈ M , r ∈ R,

(1) f (x + y) = f (x) + f (y),(2) f (r x) = r f (x).If a (module) homomorphism f : M → N is injective, then f is called a

monomorphism; if f is surjective, then f is called an epimorphism. Sometimeswe write M

� � �� N for a monomorphism from M to N and write M �� �� N for anepimorphism from M onto N . If there is an epimorphism from M onto M ′, thenwe say that M ′ is the epimorphic image of M . If f is both a monomorphism andan epimorphism, then f is called an isomorphism. In this case, we also say thatmodules M and M ′ are isomorphic, denoted by M ∼= M ′.Example 1.2.13 Let f : M → M ′ be a module homomorphism. Define

Ker( f ) = {x ∈ M | f (x) = 0}.

Then Ker( f ) is a submodule of M , which is called the kernel of f . And

Im( f ) = f (M) = { f (x) | ∀x ∈ M}

is a submodule of M ′, which is called the image of f . Define

Coker( f ) = M ′/Im( f ),

which is called the cokernel of f . Similarly to the proof of Theorem1.2.4, we canprove that f is a monomorphism if and only if Ker( f ) = 0; f is an epimorphism ifand only if Im( f ) = M ′, that is, Coker( f ) = 0.Example 1.2.14 Similarly to ring homomorphisms, we have the following state-ments for module homomorphisms:

(1) For x ∈ M , define f : M → M ′ by f (x) = 0. Then f is a homomorphism,which is called the zero homomorphism and denoted by f = 0.

(2) If f : M → M ′ is a homomorphism, then f (0) = 0 and for any x, y ∈ M ,f (x − y) = f (x) − f (y).

(3) For any module M , the identity map

1M : M → M, 1M(x) = x, x ∈ M

is an isomorphism.(4) Let N be a submodule of a module M . Then the inclusion map

λ : N → M, λ(x) = x, x ∈ N

is a monomorphism.

12 1 Basic Theory of Rings and Modules

(5) Let N be a submodule of a module M . Then the natural homomorphism

π : M → M/N , π(x) = x, x ∈ M

is an epimorphism. We call π the natural projection from M toM/N.(6) Let g : M → M ′ be a module homomorphism and let N be a submodule of

M . Then, by the natural way, g can be considered as a homomorphism from N toM ′, which is called the restriction of g on N , denoted by g|N .

(7) Let f : N → M ′ and g : M → M ′ be module homomorphisms, N asubmodule of M , and for any x ∈ N , g(x) = f (x). Then g is called an extensionof f over M . In this case, g|N = f .Example 1.2.15 Let f : A → B and g : B → C be module homomorphisms. Thenit is easy to prove:

(1) g f : A → C is also a homomorphism.(2) Ker(g f ) = f −1(Ker(g)) and Im(g f ) = g(Im( f )).

Example 1.2.16 Let {Mi | i ∈ Γ } be a family of modules. For any xk ∈ Mk , k ∈ Γ ,define λk : Mk → ⊕

iMi ⊆ ∏

iMi , such that λk(xk) is the element whose k-th

component is xk and the others are 0. Then λk is a monomorphism, which is calledthe standard embedding on the k-th component. Define pk : ∏

iMi → Mk by

pk([xi ]) = xk . Then pk is an epimorphism, which is called the projection on thek-th component.

Example 1.2.17 Let f : A1 → B1 and g : A2 → B2 be module homomorphisms.Define

h : A1 ⊕ A2 → B1 ⊕ B2, h(a1, a2) = ( f (a1), g(a2)), a1 ∈ A1, a2 ∈ A2.

Then h is also a homomorphism, which is also referred to as h = f ⊕g. Furthermore,h is a monomorphism (resp., an epimorphism, an isomorphism) if and only if bothf and g are monomorphisms (resp., epimorphisms, isomorphisms). More generally,let {Mi } and {Ni } be two families of modules and for each i , let fi : Mi → Ni be ahomomorphism. Define

h1 :⊕

i

Mi →⊕

i

Ni , h1([xi ]) = [ fi (xi )], xi ∈ Mi

andh2 :

∏

i

Mi →∏

i

Ni , h2([xi ]) = [ fi (xi )], xi ∈ Mi .

Then h1 and h2 are homomorphisms, denoted by h1 = ⊕ fi and h2 = ∏ fi ,respectively. Then h1 (resp., h2) is a monomorphism (resp., an epimorphism, anisomorphism) if and only if each fi is a monomorphism (resp., an epimorphism, anisomorphism).

1.2 Ring Homomorphisms and Module Homomorphisms 13

Similarly to the fundamental theorem on ring homomorphisms, we can obtain thefollowing results (from Theorems1.2.18 to 1.2.21).

Theorem 1.2.18 (Fundamental Theorem on Homomorphisms) Let f : M → M ′be a (module) homomorphism and let N be a submodule of M with N ⊆ Ker( f ).Write M = M/N. Then there is a unique homomorphism f , called the inducedhomomorphism of f , such that the following diagram commutes:

Mf ��

π ������

�� M′

Mf

����

�

and f is a monomorphism if and only if N = Ker( f ).Corollary 1.2.19 (First Isomorphism Theorem) Let f : M → M ′ be a (module)homomorphism. If f is an epimorphism, then M/Ker( f ) ∼= M ′.Theorem 1.2.20 (Second Isomorphism Theorem) Let H and N be submodules ofa module M. Then

H/(H ∩ N ) ∼= (H + N )/N ,

which maps x + (H ∩ N ) → x + N, x ∈ H.Theorem 1.2.21 (Third Isomorphism Theorem) Let H and N be submodules of amodule M with N ⊆ H. Then M/H ∼= (M/N )/(H/N ).Theorem 1.2.22 Let M be an R-module and let {Mi } be a family of submodules ofM, satisfying:

(1) M = ∑iMi ,

(2) Mi ∩ ∑j �=i

M j = 0 for each i .Then M ∼= ⊕

iMi . In this case, M is called the internal direct sum of {Mi }.

Proof Define ϕ : ⊕iMi → M by ϕ([xi ]) = ∑

ixi , where xi ∈ Mi . Since only a

finite number of xi is not 0, ϕ is well-defined. It is easy to see that ϕ is a modulehomomorphism.

Let ϕ([xi ]) = ∑ixi = 0. For any i , we get xi = − ∑

j �=ix j , therefore xi ∈ Mi ∩

∑j �=i

M j = 0. So [xi ] = 0. Thus ϕ is a monomorphism.For any x ∈ M , by (1), x = ∑

ixi , xi �= 0 for only a finite number of indices i .

Let y = [xi ] be an element of ⊕iMi , whose i-th component is xi and the others are

0. Then ϕ(y) = x . Thus ϕ is an epimorphism, and therefore an isomorphism. �

14 1 Basic Theory of Rings and Modules

Remark 1.2.1 If M is an internal direct sum of a family {Mi } of submodules of M ,then we write M = ⊕

iMi . In this case, Mi is called a direct summand of M .

Definition 1.2.23 Let R and T be rings and let M be both an R-module and aT -module,

t (r x) = r(t x), r ∈ R, t ∈ T, x ∈ M.

Then M is called an R-T bimodule.

Any R-module naturally can be seen as an R-R bimodule.

Example 1.2.24 Let ϕ : R → T be a ring homomorphism.(1) Let M be a T -module. Define

r x = ϕ(r)x, r ∈ R, x ∈ M.

Then M is also an R-module. Since t (r x) = tϕ(r)x = ϕ(r)t x = r(t x), M becomesan R-T bimodule.

(2) Clearly every ideal of T can be seen as an R-T bimodule.(3) Let I be an ideal of R. Define

I T = {n∑

i=1ai xi | ai ∈ I, xi ∈ T, n ∈ Z+}.

Then I T is an ideal of T .

Example 1.2.25 Let I be an ideal of R and let R = R/I . Let M be an R-module.For r ∈ R, x ∈ M , define r x = r x . Then M becomes an R-module.

1.3 Finitely Generated Modules and Matrix Methods

1.3.1 Finitely Generated Modules

Let M be an R-module and X ⊆ M . Define

(X) =⋂

{N | N is a submodule of M and X ⊆ N },

which is called the submodule generated by X . Trivially, if X = ∅, then (X) = 0.Theorem 1.3.1 Let X be a nonempty subset of an R-module M.

(1) (X) = {r1x1 + · · · + rnxn | n is a positive integer, ri ∈ R, xi ∈ X, i =1, 2, . . . , n}.

(2) If X = {x1, x2, . . . , xn} is a finite set, then

1.3 Finitely Generated Modules and Matrix Methods 15

(X) = {r1x1 + · · · + rnxn | ri ∈ R, xi ∈ X} = Rx1 + Rx2 + · · · + Rxn.

Proof (1) Let H be the right hand side of (1). Then clearly X ⊆ H and H is asubmodule of M . Thus (X) ⊆ H . Conversely, let T be a submodule of M whichcontains X . Then for any x = ∑

j

16 1 Basic Theory of Rings and Modules

Example 1.3.6 (1) For any ring R, consider the natural ring homomorphism ϕ :Z → R given by ϕ(m) = m1, where m ∈ Z and 1 is the identity element in R. ThusKer(ϕ) = (n) since Z is a PID. It is reasonable that we may assume n � 0. Such anonnegative integer n is called the characteristic of R.

(2) If R is a domain, then the characteristic of R is either 0 or a prime number p.(3) The characteristic of Z is 0; if n is a positive integer, then the characteristic of

Zn := Z/(n) is n.(4) If the characteristic of R is n, then Zn is isomorphic to a subring of R. We

regard Zn as a subring of R, which is the smallest subring of R and is called theprime subring of R. Thus if the characteristic of R is 0, then the prime subring ofR is Z and if the characteristic of R is a positive integer n, then the prime subring ofR is Zn .

1.3.2 Simple Modules, Maximal Submodules, and Zorn’sLemma

Definition 1.3.7 Let M be a nonzero R-module. Then M is said to be simple if theonly submodules of M are trivial.

Definition 1.3.8 Let M be an R-module and let N be a proper submodule of M .Then N is said to bemaximal if N is a maximal member, with respect to inclusion,of the set of proper submodules of M . In other words, a submodule N of M ismaximal if and only if (i) N is a proper submodule of M and (ii) there does not exista submodule A of M with N ⊂ A ⊂ M . In particular, a maximal submodule of R iscalled a maximal ideal of R.

Example 1.3.9 (1) Let M be an R-module. Then M is simple if and only if M = Rxfor any 0 �= x ∈ M .

(2) Let A be a proper submodule of a module M . Then A is a maximal submoduleof M if and only ifM/A is simple.

(3) M = Rx is a simple module if and only if ann(x) is a maximal ideal of R.In order to prove that every finitely generated module has a maximal submodule,

in particular, every ring R has a maximal ideal of R, we need Zorn’s lemma inset theory. In fact, it will be seen that many proofs of conclusions in this book areobtained by using Zorn’s lemma.

Let P be a nonempty set. If � is a binary relation over P satisfying:(1) For any x ∈ P , x � x ;(2) if x, y, z ∈ P with x � y and y � z, then x � z;(3) if x, y ∈ P with x � y and y � x , then x = y,

then (P,�) is called a partially ordered set (or simply, a poset). A partially orderedset (P,�) is said to be totally ordered if for each x, y ∈ P , it is the case that atleast one of x � y, y � x holds.

1.3 Finitely Generated Modules and Matrix Methods 17

It is easy to see that every nonempty subset of a partially (resp., totally) orderedset is also partially (resp., totally) ordered. We usually call a totally ordered subsetof a partially ordered set a chain.

Definition 1.3.10 Let M be any R-module.(1) A nonempty set consisting of some submodules ofM forms a partially ordered

set under inclusion relation. A totally ordered subset of such a partially ordered setis called a chain of submodules. In particular, a totally ordered set of ideals of R iscalled a chain of ideals of R.

(2) Let {Ai | i ∈ Γ } be a family of submodules of M . Then {Ai | i ∈ Γ } iscalled a directed set if for any i, j ∈ Γ , there exists k ∈ Γ such that Ai ⊆ Ak andA j ⊆ Ak . For example, a chain of submodules is a directed set of submodules.Example 1.3.11 Let M be an R-module.

(1) Let {Ai | i ∈ Γ } be a directed set of submodules of M . Then A = ⋃iAi is a

submodule of M , which is called a direct union of a family {Ai } of submodules.(2)Note thatM = ⋃ N , where N ranges over all finitely generated submodules of

M . Thus every module M is a direct union of all finitely generated submodules of M .

Let P be a partially ordered set and let a ∈ P . Let B be a subset of P . Then a iscalled an upper bound (resp., a lower bound) of B if for any x ∈ B, we have x � a(resp., x � a). Note that an upper bound (resp., a lower bound) a is not necessarilyin B. If for x ∈ P , whenever a � x (resp., a � x), we have x = a, then a is calleda maximal element (resp., minimal element) of P . Note that a maximal element(resp., minimal element) does not necessarily exist and is not unique when it exists.If for any x ∈ P , we have x � a (resp., x � a), then a is called the greatest element(resp., smallest element) of P . The greatest element (resp., smallest element) isunique if it exists.

Theorem 1.3.12 (Zorn’s Lemma) Let P be a partially ordered set which has theproperty that every nonempty totally ordered subset of P has an upper bound in P.Then P has at least one maximal element.

We accept without proof that Zorn’s lemma and the axiom of choice are equivalent(i.e., each of them can be deduced from the other).

Theorem 1.3.13 Let M be a finitely generated R-module and let N be a propersubmodule of M. Then there exists a maximal submodule A of M such that N ⊆ A.Therefore every finitely generated module has a maximal submodule. In particular,every ring has a maximal ideal of R.

Proof Let {x1, x2, . . . , xn} be a generating set of M . Then M = Rx1 + Rx2 + · · · +Rxn . Set

Γ = {A | A is a proper submodule ofM and N ⊆ A}.

Since N ∈ Γ , we know that Γ is nonempty.

18 1 Basic Theory of Rings and Modules

Let {Ai } be a chain in Γ . By Example 1.3.11, A = ⋃iAi is a submodule of M .

If A = M , then for i = 1, . . . , n, there exists ki such that xi ∈ Aki . Since {Ai } is achain, there exists k such that for all i = 1, . . . , n, we have xi ∈ Ak . Thus Ak = M ,a contradiction. Therefore A �= M , and thus there exists an upper bound A ∈ Γ forthe chain {Ai }. By Zorn’s lemma, Γ has a maximal element A. It is clear that A is amaximal submodule of M . �

1.3.3 Jacobson Radical of a Ring

Let R be a ring. We denote by Max(R) the set of all maximal ideals of R.

Definition 1.3.14 Let R be a ring. We define the Jacobson radical of R, sometimesdenoted by J (R), to be the intersection of all the maximal ideals of R. That is,

J (R) =⋂

{m | m is a maximal ideal of R} =⋂

m∈Max(R)m.

Theorem 1.3.15 J (R) = {a ∈ R | for any x ∈ R, 1 + xa is a unit}.Proof Let J := {a ∈ R | for any x ∈ R, 1 + xa is a unit}. If a ∈ J (R), then forany maximal ideal m and any x ∈ R, we have xa ∈ m. If 1 + xa is not a unit,then R(1 + xa) is a proper ideal of R. Thus there is a maximal ideal m0 such that1 + xa ∈ m0. Therefore 1 = (1 + xa) − xa ∈ m0, which contradicts the fact thatm0 is a proper ideal of R. Thus 1 + xa is a unit. Therefore a ∈ J .

Conversely, let a ∈ J . If a /∈ J (R), there exists a maximal ideal m such thata /∈ m. Since m is a maximal ideal of R, we have R = m + Ra. Thus 1 = y − xa,y ∈ m, x ∈ R. So y = 1+ xa is a unit, which contradicts the fact that m is a properideal of R. Thus a ∈ J (R). Therefore J (R) = J . �Corollary 1.3.16 Let I be an ideal of a ring R. Then I ⊆ J (R) if and only if1 + I = {1 + a | a ∈ I } ⊆ U (R).

1.3.4 Matrix Methods

Discussions of rings and modules often use matrices over rings. Let ri j ∈ R, i =1, . . . ,m, j = 1, . . . , n. We call

A =

⎛

⎜⎜⎜⎝

r11 r12 · · · r1nr21 r22 · · · r2n...

.... . .

...

rm1 rm2 · · · rmn

⎞

⎟⎟⎟⎠

1.3 Finitely Generated Modules and Matrix Methods 19

an m × n matrix over R. An n × n matrix is known as a square matrix of order n.We denote by E or En the identity matrix of order n. By the usual method, we candefine addition and multiplication of matrices.

Let I be an ideal of R and let R = R/I . Write

A =

⎛

⎜⎜⎜⎝

r11 r12 · · · r1nr21 r22 · · · r2n...

.... . .

...

rm1 rm2 · · · rmn

⎞

⎟⎟⎟⎠

Trivially, we have

A + B = A + B, AB = A B.

Example 1.3.17 Let {x1, x2, . . . , xn} be a generating set of M . Then for any y1, . . . ,ym ∈ M , we can write

yi = ri1x1 + ri2x2 + · · · + rinxn, ri j ∈ R.

Set A = (ri j ). Then ⎛⎜⎝y1...

ym

⎞

⎟⎠ = A⎛

⎜⎝x1...

xn

⎞

⎟⎠

Let A be a square matrix of order n. Then A is said to be invertible if there exists asquare matrix B of order n such that AB = En and BA = En . In order to determinewhether A is invertible, we introduce the concept of the determinant of a squarematrix over a ring. For A = (ai j ) a square matrix of order n, define

det(A) =∑

τ

(−1)τa1i1a2i2 · · · anin ,

which is called the determinant of thematrix A, where τ = i1i2 · · · in is a permutationof a set {1, 2, . . . , n} and

(−1)τ ={

1, if τ is an even permutation,

−1, if τ is an odd permutation.

For the product of matrices,

det(AB) = det(A) det(B).

20 1 Basic Theory of Rings and Modules

The (i, j)minor of A, denoted by Mi j , is the determinant of the (n−1)× (n−1)matrix that results from deleting row i and column j of A.

The cofactor matrix of A is the n × n matrix C whose (i, j) entry is the (i, j)cofactor of A,

Ci j = (−1)i+ j Mi j .

The adjoint of A, denoted by adj(A), is the transpose of C , that is, the n × nmatrix whose (i, j) entry is the ( j, i) cofactor of A,

adj(A)i j = C ji = (−1)i+ j M ji .

Then it is easy to see that

(adj(A))A = A(adj(A)) = det(A)E .

Theorem 1.3.18 Let A be a square matrix of order n.(1) A is invertible if and only if det(A) is a unit of R. In this case, the equation

above yields:

adj(A) = det(A)A−1 and A−1 = 1det(A)

adj(A).

(2) Let I be an ideal of R, I ⊆ J (R), and R = R/I . If A is invertible, then A isinvertible.

Proof (1) This is trivial. (2) Since det(A) = det(A) is a unit of R, by Exercise 1.14,det(A) is a unit of R. Therefore A is invertible. �Theorem 1.3.19 Let M be a finitely generated R-module and let I be an ideal of R.

(1) If M = I M, then there exists a ∈ I such that (1 − a)M = 0. Furthermore,I + ann(M) = R.

(2) Let x ∈ R. If xM ⊆ I M, then there exists a positive integer n and a ∈ I suchthat (xn − a)M = 0.Proof We only prove (1), since the proof of (2) is similar to that of (1). Let{x1, . . . , xn} be a generating set of M . Since M = I M , for each i we can write

xi =n∑

j=1ai j x j , ai j ∈ I.

Set A = (ai j ). Then⎛

⎜⎜⎜⎝

x1x2...

xn

⎞

⎟⎟⎟⎠ = A

⎛

⎜⎜⎜⎝

x1x2...

xn

⎞

⎟⎟⎟⎠ or (E − A)

⎛

⎜⎜⎜⎝

x1x2...

xn

⎞

⎟⎟⎟⎠ = 0.

1.3 Finitely Generated Modules and Matrix Methods 21

It is trivial that det(E − A) = 1 − a, where a ∈ I . Multiplying both sides byadj(E − A), we have (1 − a)M = 0. Since 1 = a + (1 − a), it follows thatI + ann(M) = R. �Theorem 1.3.20 (Nakayama’s Lemma) Let I be an ideal of R, I ⊆ J (R), and Mbe a finitely generated module. If M = I M, then M = 0.Proof By Theorem1.3.19, there exists a ∈ I such that (1 − a)M = 0. Since I ⊆J (R), it follows that 1 − a is a unit. Therefore, M = 0. �

Let I be an ideal of a ring R. Then I is said to be nilpotent if there exists a positiveinteger n such that I n = 0.Theorem 1.3.21 Let I be an ideal of R, I ⊆ J (R), and N be a submodule of anR-module M.

(1) If M/N is a finitely generated and M/N = I (M/N ), then N = M.(2) If M is finitely generated and N + I M = M, then N = M.(3) If I is a nilpotent ideal and N + I M = M, then N = M. Therefore, if M/IM

is finitely generated, then M is finitely generated.

Proof (1) and (2) follow from Theorem1.3.20.(3) Since N + I M = M , we have M/N = I (M/N ). If I n = 0 for some positive

integer n, then M/N = I n(M/N ) = 0. Therefore N = M . �Corollary 1.3.22 Let I be an ideal of a ring R with I ⊆ J (R) and let M be an R-module. Let x1, . . . , xn ∈ M. If M is finitely generated or I is a nilpotent ideal of R,then {x1, . . . , xn} is a generating set of M if and only if {x1, . . . , xn} is a generatingset of M/IM.

Proof Let N = Rx1 + · · · + Rxn . If {x1, . . . , xn} is a generating set of M/IM andN + I M = M , then by Theorem1.3.21, N = M . �Corollary 1.3.23 Let I be an ideal of a ring R, I ⊆ J (R), and M be a finitelygenerated R-module. Consider the following commutative diagram with exact row:

Pf

����

����

h��

Mπ �� M/I M �� 0

If h is an epimorphism, then f is an epimorphism.

Proof By the hypothesis, ( f (P) + I M)/I M = h(P) = M/I M . Thus, f (P) +I M = M . By Corollary 1.3.22, f (P) = M . Therefore, f is an epimorphism. �

22 1 Basic Theory of Rings and Modules

1.4 Prime Ideals and Nil Radical

1.4.1 Prime Ideals

The concept of prime numbers is the most important in number theory. Borrowingthe idea of prime numbers, we can define a prime ideal of a commutative ring.

Definition 1.4.1 A proper ideal p of R is said to be prime if for a, b ∈ R, wheneverab ∈ p, then a ∈ p or b ∈ p.Example 1.4.2 In the ring Z of integers, an ideal (p) (p > 0) is prime if and only ifp is a prime number.

Note that every maximal ideal is certainly prime, but every prime ideal need notbe maximal. For example, in the integral domain Z, the ideal 0 is prime, but notmaximal.

Theorem 1.4.3 (Prime Avoidance Theorem) Let I, p1, p2 . . . , pn be ideals of R andsuppose that at most two of the pi are not prime. If

I ⊆ p1 ∪ p2 ∪ · · · ∪ pn, (1.4.1)

then there exists i , 1 � i � n, such that I ⊆ pi .

Proof Throwing away superfluous p′s, we assume that I ⊆n⋃

i=1pi , but I �⊆ ⋃

j �=ip j

for i = 1, . . . , n. We will show that n = 1 in this case.If n > 1, then since I �⊆ ⋃

j �=ip j , we can choose xi ∈ I\ ⋃

j �=ip j . Since I ⊆

n⋃j=1

p j ,

we know that xi ∈ pi . Consider an element x = xn + x1x2 · · · xn−1. Then x ∈ I .Also there exists an index i such that x ∈ pi . If n = 2, then x = x1 + x2, x1 ∈ p1\p2,x2 ∈ p2\p1. If x ∈ p1, then x2 = x − x1 ∈ p1, which is impossible. By the sameargument, it is impossible that x ∈ p2. Therefore n �= 2. If n > 2, by the hypothesis,we can suppose that p3, . . . , pn all are prime ideals. If x ∈ pn , then x1x2 · · · xn−1 ∈ pn .Since pn is a prime ideal, there exists i < n such that xi ∈ pn . This contradicts thechoice of xi . Now suppose that x ∈ pk for k < n. Since x1x2 · · · xn−1 ∈ pk , wehave xn ∈ pk , which contradicts the choice of xn . Thus it is impossible that n > 2.Therefore n = 1. �Theorem 1.4.4 Let Γ be the set of non-finitely generated ideals of R. If p is amaximal element of Γ , then p is a prime ideal.

Proof Let a, b ∈ R such that ab ∈ p. If a /∈ p and b /∈ p, then set J = {r ∈ R |ra ∈ p}. Thus b ∈ J , p ⊂ p + Ra, and p ⊂ J . Therefore, both p + Ra and J arefinitely generated. Let {p1 +r1a, . . . , pn +rna} be a generating set of p+ Ra, wherep1, . . . , pn ∈ p, r1, . . . , rn ∈ R and let {x1, . . . , xm} be a generating set of J .

1.4 Prime Ideals and Nil Radical 23

For any x ∈ p, we have x ∈ p + Ra. Therefore, we can write

x =n∑

i=1ci (pi + ria) =

n∑

i=1ci pi + (

n∑

i=1ciri )a,

where ci ∈ R. Let s =n∑

i=1ciri . Then sa = x −

n∑i=1

ci pi ∈ p. Therefore s ∈ J . We

also have s =m∑j=1

d j x j , where d j ∈ R. Therefore

x =n∑

i=1ci pi +

m∑

j=1d j x ja.

Now we can deduce that {pi , x ja | i = 1, . . . , n, j = 1, . . . ,m} is a generating setof p. Therefore p is finitely generated. This contradicts the fact that p ∈ Γ . Therefore,p is a prime ideal. �

Definition 1.4.5 Let S be a nonempty subset of a ring R. Then S is said to bemultiplicatively closed if it satisfies: (1) 1 ∈ S; (2) if s1, s2 ∈ S, then s1s2 ∈ S. Weoften call it amultiplicative subset.

Example 1.4.6 Let R be a ring.(1) If a ∈ R\{0}, then S = {an | n is a nonnegative integer} is a multiplicative

subset.(2)Let S be the set of all non-zero-divisors of R. Then S is a multiplicative subset.(3) If p is a prime ideal of R, then S = R\p is a multiplicative subset. In particular,

if R is an integral domain and set S = R\{0}, then S is a multiplicative subset.(4) In general, if {pi | i ∈ Γ } is a family of prime ideals of R, then S = R\(⋃

ipi )

is a multiplicative subset.

Theorem 1.4.7 Let I be an ideal of a ring R and let S be a multiplicative subsetof R with I ∩ S = ∅. Then there exists a prime ideal p of R such that I ⊆ p andp ∩ S = ∅.Proof Let Γ = {J | J is an ideal of R, I ⊆ J, J ∩ S = ∅}. Since I ∈ Γ , then Γ isnonempty. It is easy to see that Γ becomes a partially ordered set under set inclusion.Let {Ji } be a chain of ideals in Γ . Then it is easy to see that J = ⋃

iJi is an ideal of

R which contains I . If s ∈ J ∩ S, then there exists an index i such that s ∈ Ji , whichis impossible. Thus J ∩ S = ∅, that is, J ∈ Γ . Therefore J is an upper bound of thechain {Ji } in Γ . By Zorn’s lemma, Γ has a maximal element, say p. Now we willprove that p is a prime ideal of R.

Let a, b ∈ R, ab ∈ p. If a /∈ p and b /∈ p, then p ⊂ p + Ra and p ⊂ p + Rb.By the maximality of p, there exist s1 ∈ (p + Ra) ∩ S and s2 ∈ (p + Rb) ∩ S.Write s1 = p1 + r1a, s2 = p2 + r2b, p1, p2 ∈ P , r1, r2 ∈ R. We also have

24 1 Basic Theory of Rings and Modules

s1s2 = (p1 + r1a)(p2 + r2b) = p1 p2 + r1ap2 + r2bp1 + r1r2ab ∈ p, whichcontradicts the fact that p ∩ S = ∅. Therefore, p is a prime ideal. �Definition 1.4.8 Let p be a prime ideal of R. Then p is said to beminimal if p doesnot contain other prime ideals.

Definition 1.4.9 Let I be an ideal of R and let p be a prime ideal of R containingI . Then p is called a minimal prime ideal over I if there is no other prime idealsbetween I and p.

Trivially, every minimal prime ideal of R is a minimal prime ideal over the zeroideal. In addition, p is a minimal prime ideal over an ideal I if and only if p/I is aminimal prime ideal of R/I.

Using Exercise 1.6 and Zorn’s lemma, we can prove the following:

Theorem 1.4.10 If I is an ideal of R and p is a prime ideal of R containing I , thenthere exists a minimal prime ideal q containing I such that q ⊆ p.

1.4.2 Nil Radical and Radical of an Ideal

Definition 1.4.11 (1) Let I be an ideal of R. Then I is called a nil ideal if for anya ∈ I , there is a positive integer n such that an = 0.

(2) For a ring R, set

nil(R) = {a ∈ R | an = 0 for some positive integer n}.

By Example 1.1.1, nil(R) is an ideal of R, which is called the nil radical of R.

It is trivial that every nilpotent ideal is a nil ideal, but in general a nil ideal is notnecessarily a nilpotent ideal (see Example 1.7.23).

Theorem 1.4.12 (1) If p is a prime ideal of R, then nil(R) ⊆ p.(2) (Krull)

nil(R) =⋂

{p | p is a prime ideal of R}=

⋂{p | p is a minimal prime ideal of R}.

Proof (1) If x ∈ nil(R), then there exists a positive integer n such that xn = 0 ∈ p.Since p is a prime ideal, we have x ∈ p. Therefore, nil(R) ⊆ p.

(2) Write I = ⋂{p | p is a prime ideal of R}. Then by (1), nil(R) ⊆ I . If x /∈nil(R), then for any positive integer n, xn �= 0. Since S := {xn | n � 0} is amultiplicative subset and 0 /∈ S, by Theorem1.4.7, there exists a prime ideal p suchthat p ∩ S = ∅. Thus x /∈ p, and so x /∈ I . Therefore I = nil(R).

1.4 Prime Ideals and Nil Radical 25

Set J := ⋂{p | p is a minimal prime ideal of R}. Then trivially, I ⊆ J . For thereverse inclusion, if x ∈ J , then for any prime ideal p, there exists a minimal primeideal such that q ⊆ p, Thus x ∈ q ⊆ p. Therefore x ∈ I . �Definition 1.4.13 Let I be an ideal of R. Set

√I = {x ∈ R | there exists a positive integer n such that xn ∈ I }.

Then√I is an ideal of R, which is called the radical of I . Note that I ⊆ √I . An

ideal I is called a radical ideal if√I = I .

Trivially, nil(R) = √0. By Theorem1.4.12, we have:Theorem 1.4.14 Let I be an ideal of R. Then:

(1) The radical of I is the intersection of the minimal primes over I . That is,

√I =

⋂{p | p is a prime ideal and I ⊆ p}

=⋂

{p | p is a minimal prime ideal over I }.

(2) nil(R/I ) = √I/I .Theorem 1.4.15 (1) If A is a finitely generated nil ideal of R, then A is nilpotent.

(2) Let I be an ideal of R and let A be a finitely generated ideal of R. Supposethat A ⊆ √I . Then there exists a positive integer n such that An ⊆ I .Proof (1) Let A = (a1, a2, . . . , an). In the case n = 1, trivially, A is nilpotent.Write A1 = Ra1 and A2 = (a2, . . . , an). By the induction hypothesis, we can letAk1 = Ak2 = 0 for some positive integer k. Then A2k = (A1 + A2)2k ⊆ A2k1 +A2k−11 A2 + · · · + A1A2k−12 + A2k2 = 0.

(2) This follows by applying (1) for a ring R := R/I and an ideal A = {a | a ∈ A}.Definition 1.4.16 Let R be a ring. Then R is said to be reduced if nil(R) = 0.

For any ring R, since nil(R/nil(R)) = 0, R/nil(R) is a reduced ring.

1.5 Quotient Rings and Quotient Modules

1.5.1 Local Rings

Localization is a useful method in the commutative ring theory. In this section, westudy localization methods of rings and modules. First we study local rings.

Definition 1.5.1 A ring R is said to be local if R has the only one maximal ideal.

26 1 Basic Theory of Rings and Modules

We often use (R,m) to denote that R is a local ring with the maximal ideal m.Obviously any field is a local ring. If (R,m) is a local ring, then J (R) = m.Example 1.5.2 Let p ∈ Z be a prime number. Then for any n � 1, the ring R =Z/(pn) is a local ring.

Theorem 1.5.3 A ring R is local if and only if the set of nonunit elements forms anideal.

Proof Let m denote the set of nonunit elements of R. Suppose that R is a local ringwith the maximal ideal M . Since M is a proper ideal of R, each element of M is nota unit. Thus M ⊆ m. If x ∈ m, then x is not a unit. So (x) is a proper ideal of R.Then x ∈ (x) ⊆ M . Thus m ⊆ M . Therefore m = M is an ideal of R.

Conversely, suppose thatm forms an ideal of R. Thenm is a proper ideal. For anyproper ideal I of R, any element of I is a nonunit. So I ⊆ m. Thus m is the uniquemaximal ideal. Therefore, R is local. �

Corollary 1.5.4 A ring R is local if and