1. 2 some exceptional properties: 1.every prime ideal is contained in a unique maximal ideal. 2. sum...
TRANSCRIPT
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