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 pseudocom pct
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?
Elements of C (X ), Ideals of C (X )
• X is connected ⟺ The only idempotents
of C(X) are constant functions 0 and 1.
6
f∈C(X) is zerodivisor⟺ int Z(f ) ≠ϕ
Every element of C(X) is zerodivisor ⟺ X is an
almost P-space
Problem. 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.
| S |  |C ( S )|  |C ( X )|  |C ( D )|  2|D |
12
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 zideals, 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
Z ( f )   x X : f ( x )  0 
Z [ I ]   Z ( f ): f  I 
E. Hewitt, Rings of real-valued continuous unctions, I,
Trans. Amer. Math. Soc. 4(1948), 54-99
 1 [ F ]  f C ( X ) : Z ( f )F
Z


I  Z  1 [ Z [ I ]]
I  Z  1[ Z [ I ]]  I is a z -id eal
Z ( f )  Z ( g ), f  I , g C ( X )  g  I
Z ( f ) Z [ I ]  f  I
27
Ex. 4B.
N ecessary an d su fficien t alg eb raic co n d i tio n :
I is a z -id eal iff g iven f , if th ere ex ists g  I su ch th at f
b elo n g s to every m ax im al id eal co n tain in g g , th en f  I
M
M
f

f

f M
M
g  C(X ) : Z ( f ) 
Z (g)

A lg eb raic d efin itio n :
A n id eal I is a z-id eal
if M f  I ,  f  I .
I is a z -ideal  I  
f I
M
f
(Azarpanah-M oham adian )
28
1- Every ideal in C(X) is a z-ideal
2- C(X) is a regular ring
3- 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
Z( f 3)  Z( f )  f
1
3
1
(f) f
3
 g f
2
2
 f 3 (1  f 3 g )  Z ( f )
Z (1  f 3 g )  X ,
2
Z( f )
Z (1  f 3 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
I z  the sm allest z -ideal containing I
I  the largest z -ideal contained in I
z
Iz  
f I
M
I 
z
f
M
f
I
M
f
I z   g  C ( X ): Z ( f )  Z ( g ), f  I 
I z   g  C ( X ): Z ( g )  Z ( f )  f  I 
P z and Pz are prim e ideals
F. A zarpanah and R . M oham adian
z  ideals and

z  ideal,
in C ( X ), A cta M ath. S in. 23(6)(2007 ), 15  25.
31
C losed ideals in C ( X ) w ith m -topology are z -ideals
but not conversely, e.g. O 0 in C ( R ) (2N (7) in[G J]).
[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.
E very closed ideal is an intersection of m axim al ideals,
i.e., every closed ideal is of the form M A , w here A   X .
Question: Is the sum of every two closed
ideals in C(X) a closed ideal?
M AM B M A
B
32
W e call an ideal I a
An ideal I in C ( X ) is a
z  ideal if
I is a z-ideal.
z  ideal iff I is a z -ideal.
.
P roblem Investigate reduced rings in w hich
eve ry
z  ideal is a z -ideal.
33
Relative z-ideals
rez-ideals
D e f. For every tw o ideals I  J in C ( X ), w e call I a z -ideal
J
if Z ( f )  Z ( g ), f  I and g  J im ply that g  I .
D ef. A n ideal I in C ( X ) is called a relative z -ideal if there
exists an ideal J such that I Ø J and I is a z J -ideal.
F. Azarpanah and A. Taherifar, Relative z-ideals in C(X),
Topology Appl. 156(2009), 1711-1717.
Fact. A n ideal I in C ( X ) is a z J -ideal  I z
 K
J I
J  I for som e z -ideal K .
So relative z-ideals are also bridges
34
(a) E very p rin cip al id eal in C ( X ) is a rel ative
z -id eal iff X is an alm o st P -sp ace.
(b ) S u m o f every tw o rez -id eals is a rez -id eal
iff X is a P -sp ace.
(c) F o r every id eal J in C ( X ), su m o f ev ery tw o
z J -id eals is a z J -id eal iff X is an F -s p ace.
z   ideals ( d -ideals)
[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
D ef . A n id eal I in C ( X ) is called a z  -id eal if g  C ( X ),
f  I , an d in t X Z ( f )  in t X Z ( g ) im p ly th at g  I .
Fact. int X Z ( f )  int X Z ( g )  A nn ( f )  A nn ( g ).
Fact . int X Z ( f )    f is a zerodivisor.
P roble m : Let X be a m etric space and A and B be tw o closed sets
in X . If ( A B )   , then either A    o r B    .
Solution . A  Z ( f ), B  Z ( g ).
(Z ( f )
Z ( g ))    ( Z ( fg ))    fg is a zerodivizor
fgh  0 for som e h  C ( X )  gh  0 or f ( gh )  0.
S o e ith e r f is a z e ro d iv is o r o r g is , i.e .
A   ( Z ( g ))   o r B  = ( Z ( f ))   .
36
 f C ( X ) , P f 
P f
f  P  M in ( C ( X ))
 g C ( X ): in t X Z ( f ) 
P
in t X Z ( g )
  g C ( X ): A nn(f)  A nn(g)
Fact. T he follow ings are equivalent:
1. I is a z  -ideal.
2. If f  I , g  C ( X ), and A nn( f )  A nn( g ), then g  I .
3.  f  I , P f  I .
4.  f  I , A nn(A nn( f ))  I .

So z -ideals are also Bridges
37
S in c e
M
f
  g C ( X ): Z ( f )  Z ( g ) 
 g C ( X ):
in t X Z ( f )  in t X Z ( g )  P f ,
th e n e v e ry z   id e a l in C ( X ) is a z -id e a l.
T he set of basic z   ideals in C ( X )=  P f : f C ( X ) 
E very m em ber of a proper z   ideal in C ( X ) is zerodivisor.
[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
1 . E v e ry b a s ic z  -id e a l in C ( X ) is p rin c ip a l iff
X is b a s ic a lly d is c o n n e c te .

2 . E v e ry in te rs e c tio n o f b a s ic z -id e a l in C ( X )
is p rin c ip a l iff X is e x te re m e lly d is c o n n e c te d .
3 . E v e ry id e a l in C ( X ) c o n s is tin g e n tire ly z e ro d iv is o rs is a -id e a l iff X is a P -s p a c e .

4 . E v e ry z -id e a l in C ( X ) is a z -id e a l iff X i s a n
a lm o s t P -s p a c e .
S u m o f tw o n o n reg u lar id eals

S u m o f tw o z -id eals
M
[ 0 ,1]
  f  C ( R ):[0,1]  Z ( f )
M
[ 0 ,1]
  f  C ( R ):[  1,0]  Z ( f )
M
[ 0 ,1]
M
[ 0 ,1 ]
 M
0
39
[C . B . H uijsm ans and B . D eP agter]

Fac t . T he sum of tw o z -ideals in C ( X ) is a z -idea l
if and only if X is a quasi F -space (a spa ce X for
w hich every regular finitely generated ideal in C ( X ) is princ i pal)

I   the sm allest z -ideal containing I


I  the largest z -ideal contained in I
I 

f I
Pf
I  I z  I 
I  
P f I
Pf
I  Iz  I

[F. Azarpanah and R. Mohamadian]
40
Let X be a quasi space:
1. If I is a z -ideal and Q is a prim ary ide al in C ( X )
w hich are not in a chain, then I + Q is a pr im e z -ideal.
2. E very prim e ideal m inim al over a z -ide al is a z -ide2al
and the converse is also true in the context of C ( X ).
Questions:

W hen is every nonregular z -ideal a z -ideal?

W hen is every nonregular prim e ideal a z -ideal?
W hen is every nonregular prim e z -ideal a z  -ideal?
T h . E very nonregular prim e ideal in C ( X ) is a z  -ideal if
and only if X is a  -space (i.e., the bou ndary of each
zeroset in X is contained in a zeroset w i th em pty iterior) .
[3] F. Azarpanah and M. Karavan, On nonregular ideals and z0–
ideals in C(X), Cech. Math. J. 55(130)(2005), 397-407.
41
Essential (large) ideals
Uniform (Minimal) ideals
The Socle of C(X)
E is essential  E
U is uniform  I
m is minimal 
I  (0)
J  (0)  I , J  U
( I  m  I  (0) or I  m )
Socle of R = S(R) = Intersection of essential ideals
= Sum of uniform ideals of R
42
Q uestions: W hich of the ideals O x , P , ( f ) and the
free ideal I in C ( X ) is an e ssential ideal?
T h.
(a) A n id eal E in C ( X ) is essen tial iff
Z [E ] 
f E
Z ( f ) h as em p ty iterio r.
(b ) A n id e a l U in C ( X ) is u n ifo rm iff
it is m in im a l iff it is o f th e fo rm
m x   f  C ( X ) : X \{ x}  Z ( f )
fo r so m e iso la te d p o in t x X .
(c) T he socle of C ( X ) is
C F ( X )   f  C ( X ) : X \ Z ( f ) is finite 
43
44
* When is the socle of C(X) an essential ideal?
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.
A rin g R h as a fin ite G o ld ie d im en sio n ,
if th ere is an in teg er n  0 su ch th at a
d irect su m o f n o n zero id eals in R h as
alw ays m term s, w h ere m  n an d th ere
is a d irect su m o f u n ifo rm id eals
(w ith n te rm s) w h ich is essen tial in R
45
A set
 B i i I o f n o n zero id eals in
a rin g R is said to b e in d ep en d en t if
 i  j I B j   .
B
i
G d im ( R ) is th e sm allest card in al n u m b er
 su ch th at every in d ep en d en t set o f
n o n zero id eals in R h as card in ality less
th an o r eq u al to 
T h.
(a) C ( X ) h as a fin ite G o ld ie
d im en sio n iff X is fin ite.
(in this case Gdim C ( X )  | X |)
(b) G dim C ( X )  c ( X )  S ( X ).
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.
T h.
(a) C K ( X ) is essen tial iff X is an
alm o st lo ca lly co m p ac t sp ace
( a sp ace in w h ich every o p en set co n tain s a co m p act n eig h b o rh o o d ).
(b ) C K ( X )  C F ( X ) iff X is p seu d o co m p a ct ( every co m p act s u b set o f X h as a fin ite in terio r ).
47

 For 0<  < 1,
(h )
is essential in
(h)
 P rim e id e a ls in
C(X )
(h)
C(X )
Z ( f )  .
iff int
X
a re e s s e n tia ls
CF (X )
Z ( f )  .
 Every z -ideal in C (X ) ( h ) is essential iff int
X
 E very prim e ideal in C (X ) ( h ) is essential iff
Z ( f ) does not contain any isolated poi n t.
# 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.
 F o r all p rim e id eals P in C ( X ),
d im C ( X ) = 1 iff X is an F -sp ace.
P
48
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.
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.
49
W. K. Nicholson, Lifting idempotents and exchange rings,
Trans. Amer. Math. Soc. 229(1977), 269-278.
R. B. Warfield, A krull-Scmidt theorem for infinite sum of
modules, Proc. Amer. Math. Soc. 22(1969), 460-465.
C(X) is clean iff C(X) is an exchange ring.
R is an ex ch an g e rin g iff fo r each a  R ,  b, c  R
su ch th at b a b  b an d c (1- a )(1- b a )  1- b a
T h.
f  C ( X ) is clean  there exists a clopen set
U in X such that f -1({1})  U  X \ Z ( f ) or
Z (1- f )  U  X \ Z ( f ).
50
E x a m p le s o f c le a n e le m e n ts in C ( X ):
id e m p o te n ts , u n its , p o s itiv e p o w e r o f
-1
c le a n e le m e n ts (( f r ) ({1} )= f
and X \ Z ( f )  X \ Z ( f

f  C(X ) :
f
r

({1} ) =  .
-1
f
2
1 f
2
1 f
 1
2
({1} )
)),
C o rresp o n d in g to an y f  C ( X ),
f
-1
is clean
2
1
1 f
2
X is strongly zero dimensional if every functionally open
cover of X has an open refinement with disjoint members.
51
X is s tro n g ly z e ro d im e n s io n a l iff fo r
e v e ry p a ir A , B o f c o m p le te ly s e p a ra te d
s u b s e ts o f X , th e re e x is ts a c lo p e n s e t U
s u c h th a t A  U  X \ B .
[1 ,  1], Q , S o reg en frey lin e, ...
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.
52
53
F. Azarpanah, When is C(X) a clean ring?
Acta Math. Hungar. 94(1-2)(2002), 53-58
T h . C K ( X ) is clean iff every n h o o d o f a p o in t
co n tain s a clo p en set co n tain in g th e p o in t.
C o ro lla ry . If X is lo cally co m p act, th en
C
K
( X ) is clean iff X is zero -d im en sio n al.
54
L z ( X )  T h e set o f all z -id eals o f C ( X )
L z ( X ) is a co h eren t n o rm al Y o sid a fram e
J. Martinez and E. R. Zenk, Yosida frames,
J. pure Appl. Algebra, 204(2006), 473-492.
I  L z ( X ) is com pact  I  M
f
for som e f  C ( X ).
I  L z ( X ) is atom  I  m x for som e isolated point x  X .
 ( X )   ( L z ( X )) an d  ( )   ( L z ( ))
d im L z ( X )  c ( X )
55
Q u estio n 1 : D o es th e eq u ality  ( L z ( X ))   0 h o ld ?
W hat about the equality of  ( R L z ( X ))   0 ?
Q u estio n 2 : W h en th e eq u ality  ( L z ( X ))   ( X ) h o ld s ?
L z ( X )  L z (Y )
 ( X )   ( Y )
L z ( X )  L z (Y ), X and Y are locally com pact,
then d ( X )  d (Y ).
56