PURITY OF THE IDEAL OF CONTINUOUS FUNCTIONS WITH PSEUDOCOMPACT SUPPORT

Let CΨ (X) be the ideal of functions with pseudocompact support and let kX be the set of all points in υX having compact neighborhoods. We show that CΨ (X) is pure if and only if βX−kX is a round subset of βX, CΨ (X) is a projective C(X)-module if and only if CΨ (X) is pure and kX is paracompact. We also show that if CΨ (X) is pure, then for each f ∈ CΨ (X) the ideal (f ) is a projective (flat) C(X)-module if and only if kX is basically disconnected (F ′-space).


Introduction.
Let X be a completely regular T 1 -space, βX the Stone-Čech compactification 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 and S(f υ ) = cl υX S(f ), where f υ is the extension of f to υX, S(f β ) = cl βX S(f * ), where and f β is its extension to βX.If I is an ideal in C(X), then cozI = f ∈I coz f .Let C K (X), C Ψ (X), and I(X) be the ideal of functions with compact support, pseudocompact support, and the intersection of all free maximal ideals of C(X), respectively.
The space X is called µ-compact if C K (X) = I(X), it is called Ψ -compact if C K (X) = C Ψ (X), and it is called η-compact if C Ψ (X) = I(X).
Let µX be the smallest µ-compact subspace of βX containing X, Ψ X the smallest Ψ -compact subspace of βX containing X, and ηX the smallest η-compact subspace of βX containing X.
The following diagram illustrates the relationships between these spaces: For more information about these spaces the reader may consult [7].
A space X is called locally pseudocompact if every point of X has a pseudocompact neighborhood (nbhd), it is called basically disconnected if for each f ∈ C(X), S(f ) is clopen in X and it is called an F -space if for each f ,g ∈ C(X) such that f g = 0, then 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 in C(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 if X is locally compact, then C K (X) is pure, Brookshear [3] and De Marco [4] studied purity and projectivity, Natsheh and Al-Ezeh [11,Theorem 2.4] characterized pure ideals in C(X) to be the ideals of the form O A , where A is a unique closed subset of βX, and Abu Osba and Al In this paper, we characterize purity of C Ψ (X) using the subspace kX, the set of all points in υX 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 each f ∈ C Ψ (X).We found that if C Ψ (X) is pure, then it is projective if and only if kX is paracompact, the principal ideal (f ) is projective (flat) if and only if kX is basically disconnected (F -space).An example is given to show that these results are false if C Ψ (X) is not pure.
The following result is well known and is used very often in this article.
Proposition 1.1.For each space X, C(X) is isomorphic to C(υX), and C Ψ (X) is isomorphic to C K (υX).
In this paper, we use the above proposition together with the results we obtained in [1] to characterize purity of the ideal C Ψ (X) using the subspace kX.

The subspace kX. For each ideal
The space kX is important in classifying some properties of X and some of its extensions and it is related to the ideal C K (υX).The following propositions and corollaries illustrate this fact.
Theorem 2.6.The space Ψ X is locally compact if and only if X is locally pseudocompact and θ(C Ψ (X)) is a round subset of βX.

Purity of C Ψ (X).
Here we characterize purity of C Ψ (X) using the subspace kX.But first we need some preliminaries.
It was proved in [11,Theorem 2.4] that an ideal I in C(X) is pure if and only if I = O A where A is a unique closed subset of βX.In fact, it was proved that A must be the set f ∈I cl βX Z(f ) = θ(I).Here we show that if the ideal O A is pure, then A need not be closed, but In the following theorem we characterize purity of the ideal C Ψ (X) using properties of the subspace kX.
Theorem 3.3.The following statements are equivalent: ( The following result will be extremely useful throughout the rest of the paper. Proof.The ideal C Ψ (X) is pure if and only if C K (υX) is pure if and only if for each f ∈ C Ψ (X), S(f υ ) ⊆ kX, see Propositions 1.1 and 3.1.
Corollary 3.5.The space Ψ X is locally compact if and only if X ⊆ kX and C Ψ (X) is pure.
Proof.The result follows from Theorems 2.6 and 3.3.

It was shown in
This raises the following question: suppose that coz C Ψ (X) = f ∈C Ψ (X) S(f ), does this imply that C Ψ (X) is pure?The following example shows that this need not be true.
which implies that S is not Ψ -compact.Hence kS = S ≠ Ψ S. Therefore, Ψ S is not locally compact.So it follows by Corollary 3.5 that C Ψ (S) is not pure although S is locally pseudocompact and coz C Ψ (S) = S = f ∈C Ψ S(f ).

Some applications.
In this section, we use the characterization obtained in Theorem 3.3 and Corollary 3.4 above for purity of the ideal C Ψ (X) to characterize when C Ψ (X) is a projective C(X)-module, when every principal ideal of C Ψ (X) is projective or flat C(X)-module, and for which spaces X and Y , the two ideals C Ψ (X) and C Ψ (Y ) are isomorphic.[12,Corollary 4.11].
For the converse, we prove that C K (υX) is isomorphic to C K (υY ), then the result follows from Proposition 1.1. Suppose Then φ is a ring homomorphism.It remains to show that φ is bijective.
To see that φ is one-to-one, suppose φ(f To see that φ is onto, let f ∈ C K (υX).Define Then g ∈ C(υY ), since ϕ(S(f )) is compact.Here again we use the purity of C Ψ (X), since we assumed that S(f ) ⊆ kX.Moreover, if g(y) ≠ 0, then Finally, note that Thus φ(g) = f and so φ is onto.Hence C K (υX) is isomorphic to C K (υY ).
Here we characterize when C Ψ (X) is a projective C(X)-module.
Theorem 4.2.The ideal C Ψ (X) is a projective C(X)-module if and only if kX is paracompact and C Ψ (X) is pure.
Proof.It was proved by Brookshear [3, Theorem 3.10] that C K (X) is a projective C(X)-module if and only if coz C K (X) is paracompact and S(f ) ⊆ coz C K (X) for each f ∈ C K (X).Our result now follows from Proposition 3.1 and Corollaries 2.4 and 3.4.
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 if S(f ) is clopen in X (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 each f ∈ C Ψ (X).
Conversely, suppose that every principal ideal of C Ψ (X) is a projective C(X)-module.
Then g ∈ C Ψ (X) and f = f g.Thus C Ψ (X) is a pure ideal.
To demonstrate basic disconnectedness, we first show that for each f ∈ C K (kX), S(f ) is clopen.Then we will use this result to show that for each k ∈ C(kX), S(k) is clopen.
Let f ∈ C K (kX).Then f can be extended to a function The compactness of S(f υ ) implies that there exists k υ ∈ C(υX) such that k υ (S(f υ )) = 0 and k υ (S(g υ )) = 1.So, k ∈ Ann(f ), and g = gk.Thus the ideal (f ) is a flat C(X)module since Ann(f ) is pure.
Conversely, suppose that the principal ideal 2 and Theorem 3.2 above.