Purity of the ideal of continuous functions with pseudocompact support

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PURITY OF THE IDEAL OF CONTINUOUS FUNCTIONS

WITH PSEUDOCOMPACT SUPPORT

EMAD A. ABU OSBA

Received 7 November 2000 and in revised form 29 March 2001

LetC_Ψ(X)be the ideal of functions with pseudocompact support and letkXbe the set of all points inυXhaving compact neighborhoods. We show thatC_Ψ(X)is pure if and only if βX−kXis a round subset ofβX,C_Ψ(X)is a projectiveC(X)-module if and only ifC_Ψ(X)is pure andkXis paracompact. We also show that ifC_Ψ(X)is pure, then for eachf∈C_Ψ(X) the ideal(f )is a projective (flat)C(X)-module if and only ifkXis basically disconnected (F_-space).

2000 Mathematics Subject Classification: 54C30, 54C40, 13C11.

1. Introduction. LetXbe a completely regularT1-space,βXthe Stone-ˇCech

com-pactification of X and υX the Hewitt realcompactification of X. Let C(X) be the ring of all continuous real-valued functions defined on X. For each f ∈ C(X), let Z(f )= {x∈X:f (x)=0}, cozf =X−Z(f ), the support off =S(f )=clXcoz(f ), andS(fυ_)=cl

υXS(f ), wherefυis the extension offtoυX,S(fβ)=clβXS(f∗), where

f∗_(x)=

        

1, f (x)≥1, f (x), −1≤f (x)≤1, −1, f (x)≤ −1,

(1.1)

andfβ_{is its extension toβX. IfIis an ideal inC(X), then cozI=}

f∈Icozf. LetCK(X),CΨ(X), andI(X)be the ideal of functions with compact support, pseudo-compact support, and the intersection of all free maximal ideals ofC(X), respectively. The spaceXis calledµ-compact ifCK(X)=I(X), it is calledΨ-compact ifCK(X)= CΨ(X), and it is calledη-compact ifCΨ(X)=I(X).

LetµXbe the smallestµ-compact subspace ofβXcontainingX,ΨX the smallest

Ψ-compact subspace ofβXcontainingX, andηXthe smallestη-compact subspace of βXcontainingX.

The following diagram illustrates the relationships between these spaces:

βX

υX

ΨX

µX ηX

X

For more information about these spaces the reader may consult [7]. For each subsetA⊆βX, letMA_{= {f∈C(X):A⊆cl}

βXZ(f )}andOA= {f∈C(X): A⊆IntβXclβXZ(f )} = {f ∈C(X):A⊆IntβXZ(fβ)}. It is well known that CK(X)= OβX−X _andC

Ψ(X)=OβX−υX=MβX−υX. A subsetAofβXis called a round subset of βXifOA_=MA_{, see [10].}

A spaceXis called locally pseudocompact if every point ofXhas a pseudocompact neighborhood (nbhd), it is called basically disconnected if for eachf∈C(X),S(f )is clopen inXand it is called anF_{-space if for eachf ,g∈C(X)such thatf g=0, then}

S(f )∩S(g)= ∅.

An ideal I of C(X)is called pure if for each f ∈I, there existsg∈I such that f=f g. It is clear that in this caseg=1 onS(f ).

For any undefined terms here the reader may consult [5].

Purity attracted the attention of a lot of people working in ring and module theories. A large class of commutative rings can be classified through the pure ideals of the ring. Purity of some ideals inC(X)was studied by many authors. Kohls [8, Theorem 4.6] called it an ideal with every element having a relative identity. Brookshear [3, page 325] proved that ifXis locally compact, thenCK(X)is pure, Brookshear [3] and De Marco [4] studied purity and projectivity, Natsheh and Al-Ezeh [11, Theorem 2.4] characterized pure ideals inC(X)to be the ideals of the formOA_{, whereAis a unique} closed subset ofβX, and Abu Osba and Al-Ezeh [1, Theorem 3.2] proved thatCK(X) is pure if and only if cozCK(X)=f∈CKS(f ).

In this paper, we characterize purity ofC_Ψ(X)using the subspacekX, the set of all points inυX having compact nbhds, then we use this characterization to study some algebraic properties of this ideal, such as projectivity, when the principal ideal (f )is projective or flat for eachf∈C_Ψ(X). We found that ifC_Ψ(X)is pure, then it is projective if and only ifkXis paracompact, the principal ideal(f )is projective (flat) if and only ifkXis basically disconnected (F_{-space). An example is given to show that}

these results are false ifC_Ψ(X)is not pure.

The following result is well known and is used very often in this article.

Proposition1.1. For each spaceX,C(X)is isomorphic toC(υX), and CΨ(X)is

isomorphic toCK(υX).

Proof. Letϕ: C(X)→C(υX) be defined such thatϕ(f )=fυ_{. Thenϕ is the} required isomorphism, see [5, Section 8.1] and [6, Theorem 2.1].

In this paper, we use the above proposition together with the results we obtained in [1] to characterize purity of the idealC_Ψ(X)using the subspacekX.

2. The subspacekX. For each idealI inC(X), defineθ(I)= {x∈βX:I⊆Mx_}. Thenθ(I)=f∈IclβXZ(f ), see [5, Exercise 70.1].

Proposition2.1(see [6, Corollary 3.3 and Theorems 3.1 and 5.3]). The following statements are equivalent for any spaceX:

(i) Xis locally pseudocompact;

(ii) X⊆kX;

(iii) ηXis locally compact;

(iv) C_Ψ(X)is not contained in any fixed maximal ideal.

Proposition2.2(see [6, Theorems 3.2, 5.1, and 5.2]). For each spaceX,

(i) kX=IntβXυX=IntβXηX=IntβXΨX; (ii) ηX=X∪kX;

(iii) ΨX−X=f∈CΨ(X)(S(fυ)−S(f )).

Proposition2.3(see [1, Theorem 2.2]). For each spaceX,cozCK(X)=IntβXX.

The following result is an easy consequence of Propositions2.2and2.3.

Corollary2.4. For each spaceX,kX=cozCK(υX)=f∈CΨ(X)υX−Z(fυ).

Corollary2.5. For each spaceX,kX=f∈CΨ(X)βX−Z(fβ).

Proof. Letf∈CΨ(X)⊆C∗(X). For eachp∈βX−υX,f∈Mp∩C∗.

Sofβ_{(p)=0 for eachp∈βX−υX. ThusZ(f}β_{)=(βX−υX)∪Z(f}υ_{), andβX−} Z(fβ_{)=υX∩(βX−Z(f}υ_))=υX−Z(fυ_).

Now,f∈CΨ(X)βX−Z(fβ)=

f∈CΨ(X)υX−Z(fυ)=

fυ_∈CK(υX)υX−Z(fυ)=kX, by

Corollary 2.4.

Theorem2.6. The spaceΨX is locally compact if and only ifXis locally pseudo-compact andθ(CΨ(X))is a round subset ofβX.

Proof. See [6, Theorem 5.4] and [10, Theorem 3.3].

3. Purity ofC_Ψ(X). Here we characterize purity ofC_Ψ(X)using the subspacekX. But first we need some preliminaries.

Proposition3.1(see [1, Theorem 3.2]). For each spaceX, the idealCK(X)is pure if and only ifcozCK(X)=f∈CK(X)S(f ).

It was proved in [11, Theorem 2.4] that an idealI in C(X)is pure if and only if I=OA_{whereAis a unique closed subset ofβX. In fact, it was proved thatAmust be} the setf∈IclβXZ(f )=θ(I). Here we show that if the idealOA is pure, thenAneed not be closed, butOA_=OclβXA_.

Theorem3.2. The idealOA_{is pure if and only ifO}A_=OclβXA_.

Proof. Suppose thatOA_{is pure andf∈O}A_{. Then there existsg∈O}A _{such that} f =f g. Sofβ_=fβ_gβ _{which implies that S(f}β_)⊆cozgβ_{. HenceA⊆IntβXZ(g}β_)⊆ Z(gβ_{)⊆βX−S(f}β_)⊆Z(fβ_{). Thus cl}

βXA⊆Z(gβ)⊆βX−S(fβ)⊆IntβXZ(fβ)which implies thatf∈OclβXA_.

Theorem3.3. The following statements are equivalent:

(1)CΨ(X)is pure; (2)C_Ψ(X)=OβX−kX_;

(3)βX−kXis a round subset ofβX.

Proof. (1)(2). C_Ψ(X) is pure if and only ifC_Ψ(X)=OβX−υX _=OclβX(βX−υX)₌

OβX−IntβXυX_=OβX−kX_{, seeProposition 2.2andTheorem 3.2above.}

(2)⇒(3). MβX−kX ⊇OβX−kX =CΨ(X)=MβX−υX ⊇MβX−kX. So βX−kX is a round subset ofβX.

(3)⇒(1).OclβX(βX−υX)_=OβX−IntβXυX_=OβX−kX_=MβX−kX_=MclβX(βX−υX)_=MβX−υX₌ OβX−υX_=C

Ψ(X). So it follows byTheorem 3.2thatCΨ(X)is pure.

The following result will be extremely useful throughout the rest of the paper.

Corollary3.4. The idealC_Ψ(X)is pure if and only if for eachf∈C_Ψ(X),S(fυ_)⊆kX.

Proof. The idealC_Ψ(X)is pure if and only ifCK(υX)is pure if and only if for each f∈C_Ψ(X),S(fυ_{)⊆kX, see Propositions1.1and3.1.}

Corollary3.5. The spaceΨXis locally compact if and only ifX⊆kXandC_Ψ(X)

is pure.

Proof. The result follows from Theorems2.6and3.3.

It was shown in [1, Theorem 3.2] that CK(X)is pure if and only if cozCK(X)=

f∈CK(X)S(f ). Now, ifCΨ(X)is pure, then it is easy to see that

cozCΨ(X)=

f∈CΨ(X)

S(f ). (3.1)

This raises the following question: suppose that cozCΨ(X)=f∈CΨ(X)S(f ), does this imply thatCΨ(X)is pure? The following example shows that this need not be true.

Example 3.6. Let W∗ =[0,ω1] be the set of all ordinals less than or equal to

the first uncountable ordinal numberω1. LetW=[0,ω1)andT∗=W∗×N∗, where N∗_{denote the one point compactificationN∪ {ω0}of the natural numbers. Lett=}

(ω1,ω0), T=T∗− {t}, letA=W× {ω0}and let B= {ω1} ×N. LetS be obtained

fromT×Nby identifyingA× {2n−1}withA× {2n}and identifyingB× {2n}with B×{2n+1}. ThenSis locally compact, sinceTis, andA∩B= ∅, see [7, Example 7.3] and [9, page 240]. LetHbe obtained fromT∗×Nby identifying(A∪ {t})× {2n−1} with(A∪ {t})× {2n}and identifying(B∪ {t})× {2n}with(B∪ {t})× {2n+1}. Now,

Hisσ-compact and so it is realcompact.His not locally compact since(ω1,ω0,n)

has no compact neighborhood for eachn∈N. SoS⊆kS⊆υS⊆H. Define f:S→R

byf (α,n,1)=1/nfor allα∈W∗,n∈Nand by zero otherwise. Then cozf=W∗× N× {1}and S(f )=T× {1}is pseudocompact, noncompact. So f ∈CΨ(S)−CK(S), which implies thatSis notΨ-compact. HencekS=S≠ΨS. Therefore,ΨSis not locally compact. So it follows by Corollary 3.5thatCΨ(S)is not pure althoughSis locally pseudocompact and cozCΨ(S)=S=f∈CΨS(f ).

whenCΨ(X)is a projectiveC(X)-module, when every principal ideal ofCΨ(X)is pro-jective or flatC(X)-module, and for which spacesXandY, the two idealsCΨ(X)and C_Ψ(Y )are isomorphic.

Theorem4.1. LetC_Ψ(X)andC_Ψ(Y )be pure ideals. ThenC_Ψ(X)is isomorphic to

CΨ(Y )if and only ifkXis homeomorphic tokY.

Proof. IfCΨ(X)is isomorphic toCΨ(Y ), thenkXis homeomorphic tokY, see [12, Corollary 4.11].

For the converse, we prove thatCK(υX)is isomorphic toCK(υY ), then the result follows fromProposition 1.1.

Supposeϕ:kX→kY is a homeomorphism. Letf∈CK(υY ), thenf1◦ϕ∈C(kX),

wheref1=f|kY. But cozf=ϕ(coz(f1◦ϕ)), which implies thatϕ−1(cozf )=coz(f1◦ϕ).

Therefore clkXcoz(f1◦ϕ)=clkXϕ−1(cozf )=ϕ−1(S(f )), sinceS(f )is contained inkYby the purity ofCΨ(Y ). Now, for eachf∈CK(υX)define

gf :υX →R bygf(x)=   

f1◦ϕ(x), x∈kX,

0, x∈υX−ϕ−1_{S(f ) .} (4.1)

Then,gf∈CK(υX), sinceS(gf)=clkXcoz(f1◦ϕ)is compact.

Define ¯ϕ:CK(υY )→CK(υX)by ¯ϕ(f )=gf. Then ¯ϕis a ring homomorphism. It remains to show that ¯ϕis bijective.

To see that ¯ϕis one-to-one, suppose ¯ϕ(f )=0. Thenf1◦ϕ(x)=0 for everyx∈kX.

But coz(f1◦ϕ)=ϕ−1(cozf ), and soϕ−1(cozf )= ∅. Thereforef=0.

To see that ¯ϕis onto, letf∈CK(υX). Define

g:υY →R byg(y)=   

f◦ϕ−1_{(y), y∈kY ,}

0, y∈υY−ϕS(f ) . (4.2)

Theng∈C(υY ), sinceϕ(S(f ))is compact. Here again we use the purity ofCΨ(X), since we assumed that S(f )⊆kX. Moreover, ifg(y)≠0, thenϕ−1_{(y)∈cozf. So}

cozg⊆ϕ(cozf ).

Thus, clkYcozg⊆clkYϕ(cozf )=ϕ(clυXcozf )=ϕ(S(f )). HenceS(g)=clkYcozg is compact. It follows thatg∈CK(υY ).

Finally, note that

¯

ϕ(g)(x)=   

g1◦ϕ(x), x∈kX,

0, x∈υX−ϕ−1_{S(g) ;}

=   

f◦ϕ−1_{◦ϕ(x), x∈kX,}

0, x∈υX−ϕ−1_{S(g) ;}

=   

f (x), x∈kX, 0, otherwise; =f (x).

(4.3)

Here we characterize whenCΨ(X)is a projectiveC(X)-module.

Theorem 4.2. The idealCΨ(X)is a projective C(X)-module if and only if kX is

paracompact andCΨ(X)is pure.

Proof. It was proved by Brookshear [3, Theorem 3.10] thatCK(X)is a projective C(X)-module if and only if cozCK(X)is paracompact andS(f )⊆cozCK(X)for each f∈CK(X). Our result now follows fromProposition 3.1and Corollaries2.4and3.4.

It was proved in [2, Lemma 2] and [3, Corollary 2.5] that the principal ideal(f )is a projective (flat)C(X)-module if and only ifS(f )is clopen inX(Ann(f )is pure). We can use this result to determine when the principal ideal(f )is a projective or a flat C(X)-module for eachf∈CΨ(X).

Theorem4.3. For eachf∈C_Ψ(X), the ideal(f )is a projectiveC(X)-module if and only ifCΨ(X)is pure andkXis basically disconnected.

Proof. Suppose that kX is basically disconnected and C_Ψ(X) is pure. Let f ∈ C_Ψ(X). ThenS(fυ_{) ⊆kX since C}

Ψ(X) is pure. Now let f1 = fυ|kX and note that clkX(kX−Z(f1))=S(fυ). Since kX is basically disconnected, S(fυ) is open inkX

and therefore it is open inυX (cf.Proposition 2.2). ThusS(f )=S(fυ_{)∩X is open} inX. Hence the ideal(f )is a projectiveC(X)-module.

Conversely, suppose that every principal ideal ofCΨ(X)is a projectiveC(X)-module. For eachf∈CΨ(X),S(f )is open inX, so define

g(x)=   

1, x∈S(f ),

0, otherwise. (4.4)

Theng∈C_Ψ(X)andf=f g. ThusC_Ψ(X)is a pure ideal.

To demonstrate basic disconnectedness, we first show that for eachf∈CK(kX), S(f )is clopen. Then we will use this result to show that for eachk∈C(kX),S(k)is clopen.

Letf∈CK(kX). Thenf can be extended to a functionF∈CK(υX)with clkX(kX− Z(f ))=S(F)which is open, sinceCΨ(X)is isomorphic toCK(υX).

Now, letk∈C(kX), and a∈clkX(kX−Z(k))⊆kX. So there exists an open set U such that ¯U is compact, anda∈U⊆_U¯_{⊆kX. There existsf ∈C(kX)such that} f (a)=1 andf (kX−U)=0. Thenf∈CK(kX).

Thusa∈(kX−Z(f ))∩clkX(kX−Z(k))⊆clkX((kX−Z(f ))∩clkX(kX−Z(k)))= clkX((kX−Z(f ))∩(kX−Z(k)))=clkX(kX−Z(f k))⊆clkX(kX−Z(k)). But clkX(kX− Z(f k))is compact, and so is clopen sincef k∈CK(kX). So, clkX(kX−Z(k))is clopen inkX. ThuskXis basically disconnected.

Theorem4.4. LetXbe a space such thatCΨ(X)is pure. Then for eachf∈CΨ(X)

the principal ideal(f )is a flatC(X)-module if and only ifkXis anF_-space.

Proof. Suppose thatkXis anF_-space,f∈C

pure andS(fυ_{)⊆kX. Now ifx∈S(f}υ_{), thenx∈clkX(kX−Z(f}

1)), which implies

thatx∉clkX(kX−Z(g1)). So there exists an open setU⊆kX such thatx∈U and

U∩(kX−Z(g1))= ∅. Thereforex∈U⊆Z(g1)⊆Z(gυ). ButUis open inυX, since

kXis. SoU∩(υX−Z(gυ_{))= ∅, which implies thatx∉S(g}υ_{). ThusS(f}υ_)∩S(gυ_{)= ∅.} The compactness ofS(fυ_{)implies that there existsk}υ_{∈C(υX)such thatk}υ_(S(fυ₎₎ =0 andkυ_(S(gυ_{))=1. So,k∈Ann(f ), andg=gk. Thus the ideal(f )is a flat} C(X)-module since Ann(f )is pure.

Conversely, suppose that the principal ideal(f )is a flatC(X)-module for eachf∈ C_Ψ(X). Letg,k∈C(kX)such thatgk=0. Supposey∈clkX(kX−Z(g))∩clkX(kX− Z(k)). There exists f1 ∈ CK(υX) such that f1(y) ≠ 0. Let f = f1|kX, then y ∈

clkX(kX−Z(f g))∩clkX(kX−Z(f k)). Define

h1(x)=

  

f g(x), x∈clkXkX−Z(f g) , 0, x∈υX−kX−Z(f g) ;

h2(x)=

  

f k(x), x∈clkXkX−Z(f k) , 0, x∈υX−kX−Z(f k) .

(4.5)

Thenh1,h2∈CK(υX), sinceS(h1)andS(h2)are compact sets. Moreover,h1h2=0.

So, there existsh1∈Ann(h2)such thath1=h1h

1. Hencey∈clkX(kX−Z(f g))=

S(h1)⊆cozh 1. Buth

1(S(h2))=0, soy∉S(h2)=clkX(kX−Z(f k)), a contradiction. Hence clkX(kX−Z(g))∩clkX(kX−Z(k))= ∅andkXis anF-space.

Example4.5. LetX=[−1,1]with all its points isolated, except forx=0 it has its usual nbhds. ThenX is regular, paracompact, and consequently realcompact. So X=υXandkX=X−{0} ⊆X.

Let

f (x)=     

x, x=_n1, n∈Z∗_,

0, otherwise. (4.6)

ThenS(f )= {1/n:n∈Z∗_{} ∪ {0}. Sof∈CΨ(X)andS(f )is not contained inkX. So}

C_Ψ(X)is not a pure ideal.

The setS(f )is not open, so the ideal(f )is not projective. Ann(f )is not pure, since the function

g(x)=     

0, x=_n1, n∈Z∗_,

x, otherwise, (4.7)

belongs to Ann(f ), but S(g)=X− {1/n:n∈Z∗_{} is not a subset of coz Ann(f ),}

since for eachh∈Ann(f ),h(0)=0. So the ideal(f )is not a flatC(X)-module. Let Y=X−{0}, thenkX=Y =kY, butCΨ(X)is not isomorphic toCΨ(Y ), since the latter is pure.

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Emad A. Abu Osba: Department of Mathematics, University of Petra, Amman961343, Jordan