1

, ( ), ( ), ( )f g C X f g C X f g C X

2

( ) ( ) :C X f C X f is bounded

( ) ( )C X C X X is pseudocompct

Some exceptional properties:1. Every prime ideal is contained in a unique

maximal ideal. 2. Sum of two prime ideals is prime.3. The prime ideals containing a given prime ideal form a chain.

3

4

For each space X, there exists a completely regular Hausdorff space Y such that C(X) C(Y).≅

5

Major Objective?

• X is connected ⟺ The only idempotents of C(X) are constant functions 0 and 1.6

Elements of C (X ), Ideals of C (X )

f∈C(X) is zerodivisor⟺ int Z(f ) ≠ϕEvery element of C(X) is zerodivisor ⟺ X is an almost P-spaceProblem. Let X be a metric space and A and B

be two closed subset of X. If (A⋃B)˚≠ϕ, then either A ˚≠ϕ or B ˚≠ϕ.7

Def. A ring R Is said to be beauty if every nonzero member of R is represented by the sum of a

zerodivisor and a nonzerodivisor (unit) element.

8

9

♠. Every member of C(X) can be written as a sum of two zerodivisors

10

Theorem. C(X) is clean iff X is strongly zero-dimensional.

11

Proof: Let X be normal.

12

| || | | ( )| | ( )| | ( )| 2DS C S C X C D

13

14

15

1. Every z-ideal is semiprime.2. Sum of z-ideals is a z-ideal.3. Sum of a prime ideal and a z-ideal is a prime z-ideal.4. Prime ideals minimal over a z-ideal are z-ideals.

5. If all prime ideals minimal over an ideal are z-ideals, then that ideal is also a z-ideal.

6. If a z-ideal contains a prime ideal, then it is a prime ideal.

16

Def. An ideal E in a ring R is called essential if it intersects every nonzero ideal nontrivially.

17

18

19

20

21

22

23

24

THANKS

25

26

z-ideals

( ) : ( ) 0Z f x X f x [ ] ( ):Z I Z f f I

1[ ] ( ) : ( )f C X Z fZ F F

1[ [ ]]I Z Z I

1[ [ ]] isa -idealI Z Z I I z

E. Hewitt, Rings of real-valued continuous unctions, I, Trans. Amer. Math. Soc. 4(1948), 54-99

( ) ( ), , ( )Z f Z g f I g C X g I

( ) [ ]Z f Z I f I 27

Ex. 4B. Necessary and sufficient algebraic condition: is a -ideal iff given , if there exists such that

belongs to every maximal ideal containing , then

I z f fg Ig f I

f Mf MM

( ) : ( ) ( )f g C X Z f Z gM

An ideal isAlgebraic

a z-ideali

definitio

, .

:

f

n

f

IM I f I

(Azarpanah-Mohamadian)is a -ideal f I fMI z I 28

1- Every ideal in C(X) is a z-ideal2- C(X) is a regular ring3- X is a P-space (Gillman-Henriksen)

Whenever X is compact, then every prime z-ideal is either minimal or maximal if and only if X is the union of a finite number one-point compactification of discrete spaces. (Henriksen, Martinez and Woods)

1 1 1

3 3 3

1 2 2

3 3 3

2

3

( ) ( ) ( )

(1 ) ( ) (1 ) ,

( ) (1 )

Z f Z f f f f g f

f f g Z f Z f g X

Z f Z f g

29

[1] C.W. Kohls, Ideals in rings of continuous functions, Fund. Math. 45(1957), 28-50[2] C.W. Kohls, Prime ideals in rings of continuous functions, Illinois J. Math. 2(1958), 505-536.[3] C.W. Kohls, Prime ideals in rings of continuous functions, II, Duke Math. J. 25(1958), 447-458.

Properties of z-ideals in C(X):Every z-ideal in C(X) is semi prime.Sum of z-ideals is a z-ideal. (Gillman, Jerison)(Rudd)Sum of two prime ideal is a prime (Kohls) z-ideal or all of C(X). (Mason)Sum of a prime ideal and a z-ideal is a prime z-ideal or all of C(X). (Mason)Prime ideals minimal over a z-ideal is a z-ideal. (Mason)If all prime ideals minimal over an ideal in C(X) are z-ideals, that ideal is also a z-ideal. (Mullero+ Azarpanah, Mohamadian) Prime ideals in C(X) containing a given prime ideal form a chain. (Kohls)If a z-iIdeal in C(X) contains a prime ideal, then it is a prime ideal. (Kohls)

30

largest in

the smallest -ideal containing

the -ideal containedzz

I z I

I z I

z ff II M zfM If

MI

and are prime idealszz

P P

( ): ( ) ( ),I g C X Z f Z g f Iz

( ): ( ) ( )zI g C X Z g Z f f I

F. Azarpanah and R. Mohamadian ideals and ideal,

in ( ), Acta Math. Sin. 23(6)(2007), 15 25.

z z

C X

31

[1] L. Gillman, M. Henriksen and M. Jerison, On a theorem of

Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. 5(1954), 447-455.

[2] T. Shirota, A class of topological spaces, Osaka Math. J. 4(1952), 23-40.

Every closed ideal is an intersection of maximal ideals,

i.e., every closed ideal is of the form , where .AM A X

Question: Is the sum of every two closed ideals in C(X) a closed ideal?

A BA BM M M

0

Closed ideals in ( ) with -topology are -ideals

but not conversely, e.g. O in ( ) (2N(7) in[GJ]).

C X m z

C R

32

An ideal I in ( ) is a ideal iff is a -ideal.C X z I z

ideal if is a z-ideal.We call an ideal a z II

Investigate reduced rings in which

eve

Problem

ry ideal is a -ideal.

.z z

33

For every two ideals in ( ), we call a -ideal

if ( ) ( ), and

De

imply that .

f. I J C X I zJ

Z f Z g f I g J g I

-ideal. An ideal in ( ) is called a relative -ideal if there

exists an ideal such that and is a Def.

J

I C X zJ I J I zØ

F. Azarpanah and A. Taherifar, Relative z-ideals in C(X), Topology Appl. 156(2009), 1711-1717.

-ideal .

An ideal in ( ) is a -ideal

for some

Fact. zJ JI IKJK I

I C X z

z

So relative z-ideals are also bridges

Relative z-ideals rez-ideals

34

(a) Every principal ideal in ( ) is a relative

-ideal iff is an almost -space.

(b) Sum of every two -ideals is a -ideal

iff is a -space.

(c) For every ideal in ( ), sum of ev

C X

z X P

rez rez

X P

J C X ery two

-ideals is a -ideal iff is an -space. J Jz z X F

ideals ( -ideals)z d [1] C.B. Huijsmans and Depagter, on z-ideals and d-ideals in Riesz spaces I, Indag. Math. 42(A83)(1980), 183-195.

[2] G. Mason, z-ideals and quotient rings of reduced rings, Math. Japon. 34(6)(1989), 941-956.

[3] S. Larson, Sum of semiprime, z and d l-ideals in class of f-rings, Proc. Amer. Math. Sco. 109(4)(1990), 895-901.

35

An ideal in ( ) is called a -ideal if ( ), , and int ( ) int ( ) imply that .Def .

X X

I C X z g C Xf I Z f Z g g I

X

X

int ( ) int ( ) Ann ( ) AFact.Fact

nn ( ). int ( ) is a zerodivisor..

XZ f Z g f gZ f f

Let be a metric space and and be two closed sets

in . If ( ) , then either

Proble

o

m

r .

: X A B

X A B A B

( is a zerodivizor0 for some ( ) 0 or ( ) 0.

( ( ) ( )) ( )) fgfgh h C X gh f ghZ f Z g Z fg

So either is a zerodivisor or is, i.e.

( or = ( .( )) ( ))

f g

A Z g B Z f

( ), (Solution ).. A Z f B Z g

36

( ( ))( ) , f f P Min C Xf C X P P

X X( ): int ( ) int ( )f g C X Z f Z gP

( ): Ann(f) Ann(g)g C X

The followings are equivalent:

1. is a -ideal.2. If , ( ), and Ann( ) Ann( ), then .3. , .

4. , Ann(Ann( )) .

Fact.

f

I zf I g C X f g g If I P I

f I f I

So -ideals are also Bridgesz37

The set of basic ideals in ( )= : ( )fz C X f C XP Every member of a proper ideal in ( ) is zerodivisor.z C X

Since ( ): ( ) ( )

( ): ( ) ( ) , int int

then every ideal in ( ) is a -ideal.

X X f

fM g C X Z f Z g

g C X Z f Z g P

z C X z

[1] F. Azarpanah, O. A. S. Karamzadeh and A. Rezaei Aliabad, On ideal consisting entirely zero divisors, Comm. Algebra, 28(2)(2000), 1061-1073.[2] G. Mason, Prime ideals and quotient of reduced rings, Math. Japon. 34(6)(1989), 941-956.[3] F. Azarpanah and M. Karavan, On nonregular ideals and z0–ideals in C(X), Cech. Math. J. 55(130)(2005), 397-407.

38

2. Every intersection of basic

1. Every basic -ideal in (

-ideal in

) is principal iff is basically disconnecte.

3. Every idea

( ) is principal iff is exteremelly disconnected.

l in (

z

z C

C X

X

X

X

C X

) consisting entirely zero-divisors is a -ideal iff is a -space.

4. Every -ideal in ( ) is a -ideal iff is an almost -space.

z C X z XP

X P

Sum of two nonregular ideals

Sum of two z -ideals

[0,1]

[0,1]

[0,1]

[0,1] 0

( ):[0,1] ( )

( ):[ 1,0] ( )

f C Z fM

f C Z fM

M MM

R

R

39

the smallest -ideal containing

the largest -ideal contained in

I z I

I z I

[C. B. Huijsmans and B. DePagter]

The sum of two -ideals in ( ) is a -ideal if and only if is a quasi -space (a space for which every regular finitely generated ideal in

Fac

( ) is princ

.

i

t C X zzX F X

C X

pal)

f

f fIPf II IP P

[F. Azarpanah and R. Mohamadian]

zI I I I I Iz

40

1. If is a -ideal and is a primary ideal in ( )

which are not in a chain, then + is a prime -ideal.

2. Every prime ideal minimal over a -ideal is a -ide2al and the converse is also true

I Q C Xz

I Q z

z z

in the context of ( ).C X

Questions:When is every nonregular -ideal a -ideal?

When is every nonregular prime ideal a -ideal?

When is every nonregular prime -ideal a -ideal?

z z

z

z z

[3] F. Azarpanah and M. Karavan, On nonregular ideals and z0–ideals in C(X), Cech. Math. J. 55(130)(2005), 397-407.

Let X be a quasi space:

Every nonregular prime ideal in ( ) is a -ideal if and only if is a -space (i.e., the boundary of each zeroset in is contained in a zeroset with empty iterior)

Th .

.

C X zX

X

41

Essential (large) idealsUniform (Minimal) ideals

The Socle of C(X)

is essential (0)E E I is uniform (0) , U I J I J U

is minimal ( (0) or )m I m I I m

Socle of R = S(R) = Intersection of essential ideals = Sum of uniform ideals of R

42

Which of the ideals , , ( ) and the

free ideal in ( ) is an e

Questions:

ssential ideal?

P fOxI C X

(a) An ideal in ( ) is essential iff

[ ] ( ) has empty iterior.

h. T

f E

E C X

Z E Z f

(b) An ideal in ( ) is uniform iff

it is minimal iff it is of the form

( ) : \{ } ( )

for some isolated point .

U C X

f C X X x Z fmxx X

(c) The socle of ( ) is

( ) ( ) : \ ( ) is finiteF

C X

X f C X X Z fC 43

44

Fact:(a) The socle of C(X) is essential iff the set of isolated points of X is dense in X.(b) Every intersection of essential ideals of C(X) is essential iff the set of isolated points of X is dense in X.

* When is the socle of C(X) an essential ideal?

A ring has a finite Goldie dimension, if there is an integer 0 such that a direct sum of nonzero ideals in has always m terms, where and there is a direct sum of uniform ideals (with n te

Rn

Rm n

rms) which is essential in R45

dim( ) is the smallest cardinal number such that every independent set of

nonzero ideals in has cardinality less than or equal to

G R

R

A set of nonzero ideals in

a ring is said to be independent if .i j I ji

Bi i IR

B B

(b) Gdim ( ) ( ) ( ).C X c X S X (in this case Gdim ( ) | |)C X X

(a) ( ) has a finite Goldie dimension iff is finite.

.C X

X

Th

46

[1] F. Azarpanah, Essential ideals in C(X), Period. Math. Hungar., 31(2)(1995), 105-112.[2] F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125(1997), 2149-2154.[3] O. A. S. Karamzadeh and M. Rostami, On intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93(1985), 73-84.

a space in which every open set contains a compact neig

every

hb

compact s

orhood

ubset

(a) ( ) is essential iff is an almost loca

(b) ( ) ( ) iff is pseudo-compa

lly compac

ct (

t space( ).

.K

K F

C X X

X X XC C

Th

of has a finite interior).X

47

X

( ) ( )

( ) ( ) For 0< <1, is essential in iff int ( ) .

h C X

h hZ f

( )

( )are essentialsPrime ideals in

C X

C XF

X Every -ideal in C(X) ( ) is essential iff int ( ) .z h Z f

in Every prime ideal C(X) ( ) is essential iff ( ) does not contain any isolated poi t.

nh

Z f

# Every factor ring of C(X) modulo a principal ideal contains a nonessential prime ideal iff X is an almost P-space with a dense set of isolated points.

For all prime ideals in ( ), dim ( ) = 1 iff is an -space. P C X

C X X FP

48

Clean elements Clean ideals

An element of a ring R is called clean if it is the sum of a unit and an idempotent.

A subset S of R is called clean if each element of S is clean.

F. Azarpanah, O. A. S. Karamzadeh and S. Rahmati, C(X) vs. C(X) modulo its socle, Colloquium Math. 111(2)(2008), 315-365.

F. Azarpanah, S. Afrooz and O. A. S. Karamzadeh, Goldie dimension of rings of fractions of C(X), submitted.

49

is an exchange ring iff for each , , such that and (1- )(1- ) 1- R a R b c R

bab b c a ba ba

C(X) is clean iff C(X) is an exchange ring.

-1

( ) is clean there exists a clopen set

in such that ({1}) \ ( ) or (1- ) \ ( ).

. f C X

U X U X Z ffZ f U X Z f

Th

R. B. Warfield, A krull-Scmidt theorem for infinite sum of modules, Proc. Amer. Math. Soc. 22(1969), 460-465.

W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229(1977), 269-278.

50

X is strongly zero dimensional if every functionally opencover of X has an open refinement with disjoint members.

2

2Corresponding to any ( ), is clean

1

ff C Xf

2

2 2

111 1

f

f f

-1 -1

-1

Examples of clean elements in ( ): idempotents, units, positive power of

clean elements (( ) ({1})= ({1})

and \ ( ) \ ( )),

( ) : ({1}) = .

C X

rf frX Z f X Z f

f C X f

51

Th. The following statements are equivalent:1. C(X) is a clean ring.2. C*(X) is a clean ring.3. The set of clean elements of C(X) is a subring.4. X is strongly zero-dimensional.5. Every zerodivisor element is clean.6. C(X) has a clean prime ideal.

is strongly zero dimensional iff forevery pair , of completely separatedsubsets of , there exists a clopen set such that \ .

XA BX U

A U X B

1[ , ], Q, Soregenfrey line, ...1

52

53

If is locally compact, then

is clean iff

Coro

is

lla

zero-dimensional.(

ry

)

.

K

X

XXC

is clean iff every nhood of a point ( )

contains a clopen set containing the p n

Th.

oi t.K XC

F. Azarpanah, When is C(X) a clean ring? Acta Math. Hungar. 94(1-2)(2002), 53-58

54

( ) The set of all -ideals of ( )X z C XzL

( ) is a coherent normal Yosida frameXzL

J. Martinez and E. R. Zenk, Yosida frames, J. pure Appl. Algebra, 204(2006), 473-492.

( ) is compact for some ( ).fzI L X I M f C X

( ) is atom for some isolated point .xzI L X I m x X

dim ( ) ( )zL X c X

( ) ( ( )) and ( ) ( ( ))z zX L X L

55

( ) ( )z zL X L Y

( ) ( ), and are locally compact,then ( ) ( ).

z zL X L Y X Yd X d Y

( ) ( )X Y

0Does the equality (Questio ( )) n 1: hold?XLz

When the equality ( ( )) ( ) holds?Question 2: X XLz

0What about the equality of ( ( )) ?XLz R

56