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SOME PROPERTIES OF THE IDEAL OF CONTINUOUS
FUNCTIONS WITH PSEUDOCOMPACT SUPPORT
E. A. ABU OSBA and H. AL-EZEH
(Received 23 April 2000 and in revised form 30 August 2000)
Abstract.LetC(X)be the ring of all continuous real-valued functions defined on a com-pletely regularT1-space. LetCΨ(X)andCK(X)be the ideal of functions with pseudocom-pact support and compseudocom-pact support, respectively. Further equivalent conditions are given to characterize when an ideal ofC(X)is aP-ideal, a concept which was originally defined and characterized by Rudd (1975). We used this new characterization to characterize whenCΨ(X)is aP-ideal, in particular we proved thatCK(X)is aP-ideal if and only if CK(X)= {f∈C(X):f=0 except on a finite set}. We also used this characterization to prove that for any idealIcontained inCΨ(X),Iis an injectiveC(X)-module if and only if cozIis finite. Finally, we showed thatCΨ(X)cannot be a proper prime ideal whileCK(X) is prime if and only ifXis an almost compact noncompact space and∞is anF-point. We give concrete examples exemplifying the concepts studied.
2000 Mathematics Subject Classification. 54C30, 54C40, 13C11.
1. Introduction. LetXbe a completely regularT1-space,βXthe Stone-Cech com-pactification ofX,υXthe Hewitt real compactification ofX,C(X)the ring of all con-tinuous real-valued functions defined on X and C∗(X)the subring of all bounded functions inC(X).
For eachf∈C(X), letZ(f )= {x∈X:f (x)=0}, cozf=X−Z(f ), and suppf= cozf .IfI is an ideal ofC(X), let cozI=f∈Icozf. For eachA⊆βXletOA= {f∈ C(X):A⊆IntβXZ(f )¯}, where ¯f is the continuous extension to βXof the bounded
function
f∗(x)=
1, f (x)≥1,
f (x), −1≤f (x)≤1, −1, f (x)≤ −1,
(1.1)
and let MA= {f ∈C(X):A⊆cl
βXZ(f )}. Forp∈X, Op =Op= {f ∈C(X):p∈
IntXZ(f )}andMp=Mp= {f∈C(X):f (p)=0}. LetCK(X)denote the ideal of
func-tions with compact support andCΨ(X)denote the ideal of functions with pseudocom-pact support in the ringC(X). It is known thatCK(X)=OβX−XandCΨ(X)=OβX−υX.
For all notation and undefined terms in this paper the reader may consult [4]. It was mentioned in [4] that ifXis aP-space, then suppfis finite for eachf∈CK(X).
toCK(X)being aP-ideal. We were able to give more equivalent conditions toP-ideals
which leads us to our result, moreover we used this new characterization forP-ideals to characterize whenCΨ(X)is an injective C(X)-module and to generalize a result which was proved earlier by Vechtomov in [10]. We also study whenCK(X)is a prime
ideal and when it is maximal.
2. P-ideals. Recall that an idealIofC(X)is called pure if for eachf∈Ithere exists g∈Isuch thatf=f g, and in this caseg=1 on suppf. The idealIis called regular if for eachf∈Ithere existsg∈Isuch thatf=f2g. A spaceXis called aP-space if everyGδ-set inX is open. A pointp∈βXis called aP-point ifOp=Mp, it is called
anF-point ifOpis prime. The spaceX is called anF-space ifOpis prime for each p∈βX.
The spaceXis aP-space if and only if every prime ideal ofC(X)is maximal if and only ifOx=Mxfor eachx∈X, see [4]. Rudd in [9] extends this concept to ideals in
rings of continuous functions.
Definition2.1(see Rudd [9]). A nonzero idealIofC(X)is called aP-ideal if every proper prime ideal ofIis maximal inI.
If X is aP-space, then every ideal of C(X)is a P-ideal. Later on we will give an example of a nonzero idealIofC(X)that is aP-ideal whileXis not aP-space.
Theorem2.2(see Rudd [9]). LetIbe a nonzero ideal ofC(X). Then the following
statements are equivalent:
(1)Iis aP-ideal.
(2)Z(f )is open for eachf∈I.
(3)Every ideal ofIis pure.
(4)Every prime ideal ofC(X)which does not containIis maximal inC(X).
Now we prove the following theorem which is the main result in this section, it gives more equivalent conditions toTheorem 2.2and it will be used to characterize when CΨ(X)is aP-ideal and when it is injective.
Theorem2.3. LetIbe a nonzero ideal ofC(X). Then the following statements are
equivalent:
(1)Iis aP-ideal.
(2)Ox=Mxfor eachx∈cozIandI⊆Oxforx∉cozI.
(3)Iis a regular ring.
(4) cozIis aP-space andIis pure.
Proof. (1)⇒(2). Letx∈cozI, thenIis not contained inMx. Hence for each prime
idealP⊆Mx,Pdoes not containI, and therefore it is maximal inC(X). ThusOx=Mx,
sinceOxis the intersection of all prime ideals contained inMx, see [4].
Now, ifx∉cozI, thenI⊆Mx. Suppose that there existsf∈I−Ox. So there exists
a prime idealP⊆Mxsuch thatf∈I−P. HencePis a maximal ideal ofC(X), and so P=Mxwhich is a contradiction, sinceI⊆Mx. Hence,I⊆Oxfor eachx∉cozI.
(3)⇒(1). Clear, since the condition implies that each ideal ofIis pure.
(2)⇒(4). The condition implies thatI is pure. Let G=∞n=1Gn, where each Gn is
an open set in cozI. Lety ∈G. For each n∈Nthere exists fn∈C(X)such that
0≤fn≤1,fn(y)=0 andfn(X−Gn)=1. Letf=
∞
n=1fn2/2n. Thenf ∈C(X)and f (y)=0. Sof∈My=Oy. Hence,y∈IntXZ(f )⊆Z(f )⊆∞n=1Gn=G. ThusGis an
open set, and therefore cozIis aP-space.
(4)⇒(2). Let y∈cozI, andf∈My. Theny∈Z(f )
cozI=∞n=1{x∈X:|f (x)|< 1/n}cozI. Hence,y∈IntXZ(f ), and sof∈Oy. Now, lety∉cozIandf∈I. Purity
ofIimplies that there existsg∈Isuch thatf=f g. So suppf⊆cozg⊆cozI. Hence y∈X−suppf⊆Z(f ). Consequently,f∈Oy.
It will be shown later in Examples3.6and3.7that the two conditions in statement (4) above are both necessary.
3. When isCΨ(X)aP-ideal? IfXis aP-space, thenCK(X)is the set of all functions
inC(X)which are eventually zero (see [4]). Now our aim is to show when does the converse of this result hold? In fact, we are going to show thatCK(X)is the set of all
functions that are eventually zero if and only ifCK(X)is aP-ideal. The point is that
we were able to find a spaceXwhich is not aP-space, whileCK(X)is aP-ideal. In fact,
one can say more, but first we will need the following lemma.
Lemma3.1. IfIis a pure ideal, thensuppf⊆cozIfor eachf∈I.
Proof. Letf ∈I, then there existsg∈I such thatf =f g,which implies that
suppf⊆cozg⊆cozI.
In the following theorem we will use the result inTheorem 2.3to characterize when CΨ(X)is aP-ideal.
Theorem3.2. LetIbe an ideal contained inCΨ(X). ThenIis aP-ideal if and only
ifcozIis discrete andIis pure.
Proof. SupposeIis aP-ideal. Then cozIis aP-space, andIis pure. So suppf⊆
cozIfor eachf∈I. Hence suppfis a pseudocompactP-space. Thus cozf is a finite open set, since a pseudocompactP-space is finite (see [4]). So it follows that for each x ∈cozf ,{x}is clopen in X. Hence, cozI is a discrete space of X. The converse is clear.
The following corollary is the main result in this paper, it gives a concrete descrip-tion of the idealCK(X)when it is aP-ideal.
Corollary3.3. The following statements are equivalent: (1)CK(X)is aP-ideal.
(2) cozCK(X)is discrete andCK(X)is pure.
(3)CK(X)= {f∈C(X):f=0except on a finite set}.
Proof. The equivalence of (1) and (2) follows fromTheorem 3.2, sinceCK(X)⊆
CΨ(X).
(1)⇒(3). Letf ∈CK(X). Then suppfis finite, since it is a compactP-space. So cozf
(3)⇒(1). Letf∈CK(X). Then cozfis finite. SoZ(f )is clopen, since cozfis. Hence, CK(X)is aP-ideal.
In [6, 8] several generalizations of real compactness are studied, from them the concept of Ψ-compactness. A spaceX is called Ψ-compact if CK(X)=CΨ(X). The
following corollary relatesΨ-compactness of the spaceXandCΨ(X)being aP-ideal.
Corollary3.4. IfCΨ(X)is aP-ideal, thenXisΨ-compact.
Proof. Iff∈CΨ(X), then suppfis finite. Hencef=0 except on a finite set. Thus
CΨ(X)⊆ {f ∈C(X):f =0 except on a finite set} ⊆CK(X)⊆CΨ(X). ThusCΨ(X)= CK(X), and soXisΨ-compact.
The following is an example of a nonP-spaceXsuch thatCK(X)is aP-ideal. This
shows that the converse of the result in [4] needs not be true.
Example 3.5. Let X be the set of all rational numbers with {0}clopen and all
other points have their usual neighborhoods. Then cozCΨ(X)= {0}is discrete and CΨ(X)= {f∈C(X):f=0 except forx=0}is pure. The spaceX is not aP-space, since{2} =∞n=1(2−1/n,2+1/n)
Xis not open.
The following two examples show that the two conditions inTheorem 2.3(4) and Theorem 3.2are both necessary.
Example3.6. LetX=[−1,1]with all points isolated and{0}has its usual neigh-borhoods. Then cozCΨ(X)=X−{0}is discrete.
Now, define
f (X)=
x, x=n1 forn∈Z∗,
0, otherwise.
(3.1)
Then suppf= {1/n:n∈Z∗}{0}is compact, butZ(f )is not open. SoCΨ(X)is not aP-ideal, although cozCΨ(X)is discrete.
Example3.7. LetRbe the space of all real numbers, thenCK(R)is a pure ideal, sinceRis locally compact (see [2]), whileCK(R)is not aP-ideal.
4. Injectivity ofCΨ(X). Recall that an idealIis called an injective ideal if it is an injectiveC(X)-module. Vechtomov in [10] mentioned that ifXis locally compact, then CK(X)is an injectiveC(X)-module if and only ifXis finite. We now use the result in
Theorem 2.3to extend the result of Vechtomov to any ideal contained inCΨ(X)for any spaceX. In fact we will show that the concept ofP-ideals and injective modules are equivalent for principal ideals contained inCΨ(X).
Theorem4.1. LetI be any ideal contained inCΨ(X). ThenI is an injectiveC(X)
-module if and only ifcozIis finite.
Proof. SupposeI is an injectiveC(X)-module, thenIis a regular ring and so it
withe2=e. So cozI=coze=suppe, sinceZ(e)is clopen (see [1]). Hence cozI is a pseudocompact discrete space. Thus cozIis finite.
Conversely, suppose cozIis finite. ThenIis aP-ideal, since cozfis clopen for each f∈I, and so every ideal ofI is az-ideal inC(X). LetJbe any ideal ofC(X), and let ϕ∈HomC(X)(J, I). Define
e1(x)=
1, x∈cozϕ(J),
0, otherwise. (4.1)
Thenϕ(J)=(e1), since it is az-ideal contained inI, and cozϕ(J)is compact. Let f1∈Jsuch thatϕ(f1)=e1. LetA=cozϕ(J)cozf1. Define
g(x)=
1, x∈A,
0, otherwise. (4.2)
Theng∈I, sinceZ(e1)⊆Z(g), andIis az-ideal. Also there existsh∈Jsuch that Z(h)=Z(g). Define
k(x)= 1
h(x), x∈A, 0, otherwise.
(4.3)
Theng=kh∈J. Moreover,f1e1=f1g. Now,e1=ϕ(f1)=ϕ(f1)e1=ϕ(f1e1)= ϕ(f1g)=gϕ(f1)∈J. So for eachf∈J,ϕ(f )=ϕ(f )e1=ϕ(f e1)=f ϕ(e1). Define
Ψ:C(X)→I, such thatΨ(f )=f ϕ(e1)for eachf∈C(X). ThenΨ∈HomC(X)(C(X), I),
andΨ|J=ϕ. ThusIis an injectiveC(X)-module.
A commutative ringR with identity is called hereditary if every ideal of R is a projectiveR-module. Brookshear in [3] showed thatC(X)is hereditary if and only if Xis finite and Vechtomov in [10] showed that ifXis locally compact, thenCK(X)is
an injectiveC(X)-module if and only ifXis finite. The following corollary gives more equivalent conditions in case thatXis locally compact.
Corollary4.2. IfXis a locally compact space, then the following statements are
equivalent:
(1)C(X)is hereditary.
(2)Xis finite.
(3)CK(X)is an injectiveC(X)-module.
(4)CΨ(X)is an injectiveC(X)-module.
The following corollary gives more equivalent conditions toTheorem 3.2and relates P-ideals and injective modules.
Corollary4.3. LetIbe an ideal contained inCΨ(X). ThenIis aP-ideal if and only
if for everyf ∈I, the principal ideal(f )is an injectiveC(X)-module.
Proof. SupposeIis aP-ideal, then for eachf∈I, cozf is finite. So it follows by
Theorem 4.1that the ideal(f )is an injectiveC(X)-module.
Example4.4. LetXbe the space defined inExample 3.5, thenCΨ(X)is an injective
C(X)-module, since cozCΨ(X)= {0}is finite.
The following example shows that there exists aP-ideal which is not injective.
Example4.5. LetX=N∗, the one point compactification of the natural numbers. LetI= {f∈C(X):f=0 except on a finite set}. ThenIis aP-ideal, sinceZ(f )is open for eachf∈I, butIis not injective since cozI=Nis an infinite set.
5. When isCΨ(X)prime? Johnson and Mandelker in [6] used some ideals inC(X) to study various generalizations of real compact spaces asµ-compactness,η -compact-ness, andΨ-compactness. A spaceX is calledµ-compact ifCK(X)=I(X), the
inter-section of all free maximal ideals inC(X), it is calledη-compact ifCΨ(X)=I(X)and it is calledΨ-compact ifCΨ(X)=CK(X). In this section we will prove thatCK(X)is
maximal if and only ifI(X)is maximal andXisµ-compact. We will show thatCΨ(X) could not be a proper prime ideal. We will also show that ifI(X)is prime, then it must be maximal.
Recall that a spaceXis called an almost compact space if|βX−X| ≤1. That is,X is either compact orβX=X{∞} =X∗, the one-point compactification ofX.
It is known that ifXis an almost compact space, then it is pseudocompact (see [4]). For more properties of this space, see [7,11].
Theorem5.1. The setCΨ(X)cannot be a proper prime ideal inC(X).
Proof. Suppose thatCΨ(X)is a proper prime ideal, then it is contained in a unique
maximal ideal. SoβX−υX is a one-point set sinceCΨ(X)=OβX−υX ⊆Mx for each x∈βX−υX. ThusυXis an almost compact space and so it is pseudocompact.
HenceυXis compact and thereforeυX=β(υX)=βX. This is a contradiction. Also, since υX is compact, it follows that CK(υX)=C(υX) which implies that CΨ(X)=C(X).
Theorem5.2. The idealI(X)is prime if and only ifXis an almost compact
non-compact space.
Proof. Suppose thatI(X)is prime, then|βX−X| =1,sinceI(X)=MβX−X⊆Mx for eachx∈βX−X. HenceXis an almost compact noncompact space.
The converse is clear.
The following corollary follows easily fromTheorem 5.2.
Corollary5.3. IfI(X)is prime, then it is maximal.
Example5.4. LetTbe the Tychonoff plank, thenβT=T∗=T∪ {t0}(see [4]). So I(T)=Mt0is maximal.
Theorem5.5. The idealCK(X)is prime if and only ifXis an almost compact
non-compact space and∞is anF-point.
Proof. IfCK(X)is prime, then|βX−X| =1. SoCK(X)=O∞is prime and∞is an F-point.
The following example shows that the two conditions in Theorem 5.5 are both necessary.
Example5.6. CK(T)=Ot0 is not prime (see [4]), althoughTis an almost compact space. On the other hand, every point inβNis anF-point butCK(N)is not prime.
Theorem5.7. The following statements are equivalent: (1)CK(X)is maximal.
(2)C(X)contains only one proper free ideal.
(3)Xis an almost compact noncompact space and∞is aP-point.
(4)Xisµ-compact andI(X)is maximal.
Proof. (1)⇒(2). Clear, sinceCK(X)is the intersection of all free ideals.
(2)⇒(3).Xis noncompact, sinceC(X)contains a proper free ideal. Also sinceCK(X)
is maximal it follows thatXis an almost compact space andCK(X)=O∞=M∞, and
so∞is aP-point.
(3)⇒(4).CK(X)=O∞=M∞=I(X), soXisµ-compact andI(X)is maximal.
(4)⇒(1). Clear.
Example5.8. LetWdenote the space of all ordinals less than the first uncountable ordinal numberω1. ThenCK(W)=Mω1 (see [4]). SoCK(W)is maximal andWis an
almost compact space.
The following is an example of a spaceXsuch thatCK(X)is prime but not maximal.
Example5.9. LetNbe the set of all natural numbers and letp∈βN−N, such that p is not aP-point. Such a point exists sinceβNis not aP-space andNis discrete. So there existsf∈C∗(N), such thatp∈clβNZ(f ), butp∉IntβNZ(f )¯ where ¯f is the continuous extension off toβN. Sincep is not aP-point, it is not isolated, so the spaceX=βN−{p}is an almost compact space. Letf1=f¯|X, thenf1∈C∗(X), since
Xis pseudocompact. Moreover,p∈clβNZ(f )⊆clβNZ(f1)=clβXZ(f1), sinceZ(f )⊆ Z(f1). Hencef1∈Mp. Also, since ¯f1|
X=f1=f¯|x, it follows that ¯f1=f¯, becauseX
is dense inβX. Sop∉IntβNZ(f )¯ =IntβNZ(f1)¯ =IntβXZ(f1)¯ . Hencef1∉Op. That is, f1∈Mp−Op. Now, sinceNis anF-space, it follows thatβNis anF-space. SoβXis
anF-space, which implies thatXis anF-space. (This is becauseY is anF-space if and only ifβYis anF-space (see [4]).) Thuspis anF-point. SoCK(X)=Opis a prime ideal.
HenceCK(X)is a prime ideal which is not maximal.
Acknowledgement. This constitutes a part of a Ph.D. thesis submitted to the
University of Jordan written by the first author under the supervision of the second author.
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E. A. Abu Osba: Department of Mathematics, University of Petra, Amman961343, Jordan
E-mail address:[email protected]
