Abstract

Gelfand-Naimark's theorem states that every commutative -algebra is isomorphic to a complex valued algebra of continuous functions over a suitable compact space. We observe that for a completely regular space , is dense--separable if and only if is -cogenerated if and only if every family of maximal ideals of with zero intersection has a subfamily with cardinal number less than and zero intersection. This gives a simple characterization of -cogenerated commutative unital -algebras via their maximal ideals.

1. Introduction

In this paper, by we always mean a commutative ring with identity. Let denote the reals or the complexes. For a completely regular (topological space) , let stand for the -algebra of continuous maps . The reader is referred to [1] for undefined terms and notations. By , we mean the Stone-Čech compactification of . We denote the ring of all bounded continuous functions by . It is well known that for every completely regular space , we have (see [1, 7.1]). This note is a continuation of [2], in which we showed that for a compact space , the following are equivalent: is dense-separable if and only if is -cogenerated if and only if is separable. Here, we will drop the compactness condition of the space and improve our main result in [2]. Furthermore we generalize our results to any regular cardinal .

Let be a regular cardinal. A set is said to be an -set if . Following Motamedi in [3], we call a ring -cogenerated if for any set of ideals of with there exists an -subset of such that and is the least regular cardinal with this property. Any left or right Artinian ring is -cogenerated. Any ring with countably many distinct ideals is -cogenerated, where is one of or . In [2], it has been observed that , , where is the Cantor perfect set and are -cogenerated. We call a ring -separable if it has the following property: if is a family of maximal ideals with , then there exists an -subset of such that . In this note -separable rings are also called separable. Every -cogenerated ring is -separable. However, the converse is not true. In [2], we give an example of a separable ring which is not -cogenerated.

The density of a space is defined as the smallest cardinal number of the form , where is a dense subset of ; this cardinal number is denoted by (see [4]). A space is called dense--separable if every dense subset of has a dense--subset , which implies that and hence are less than . Dense-separable (or in our terminologies dense--separable) spaces are of great interest. Dense-separable spaces were introduced and studied by Levy and McDowell in [5]. It is evident that every dense-separable space is separable and every second countable space is dense-separable. It is well known that the Sorgenfrey line satisfies all the countability axioms but the second (see [6], page 195, example 3). Since every dense subset of is also dense in , the Sorgenfrey line is dense-separable. In [5], it has been shown that and are dense-separable.

2. Dense--Separable Spaces

Since it is easy to observe that is -cogenerated if and only if is a finite space, and in this case is a finite direct product of , we may suppose that in our discussion is an infinite space. Before proving our results, we need two easy (but useful) lemmas. Comparing these two lemmas with [2, Lemmas 1 and 2], one may observe that these two will improve and generalize [2, Lemmas 1 and 2]. In fact they give us enough space to use the full power of Mcknight's Theorem.

Lemma 1. Let be a completely regular space; then the following hold.(1)Suppose is a subset of ; then is a dense subset of if and only if ; implies that . (2)Suppose is a subset of ; then is a dense subset of if and only if ; implies that .

Proof. Part 1.  (): let and an open set containing . We must show that , and suppose on the contrary that ; then ; by complete regularity of , there exists a function such that and , and this is a contradiction to our hypothesis.
(): since is a -space and is closed in , hence is closed in . Since , we conclude that . Hence .
Part 2. It is enough to show the necessary part: suppose that ; hence there exists and a function such that and . In as much as and are contained in disjoint zero sets, there exists a function such that and (see [1, 1.15]). This is a contradiction.

Lemma 2. For , if and only if .

Proof. Suppose that ; then by the previous lemma . Now suppose that . There is a positive unit in such that and (see [1, ]). Since and , we have . Hence and this implies that .

In the next theorem, which is the main result of this note, we have generalized [2, Theorem 3] by removing the compactness hypothesis and also replacing with an arbitrary (regular) cardinal. Since , by its very definition, is compact and whenever is compact, the earlier form of our result is just a special case of the new one. On the other hand since always exists, we can (always, i.e., for an arbitrary completely regular space ) judge when the ring is -cogenerating and also -separating by looking at .

Remark 3. Before stating our main theorem, we need some useful facts. Let be a completely space and . Suppose that and . Lemma 1 (Lemma 2, resp.) shows if (, resp.), then is dense in and vice versa. Dietrich Jr. in [7] has shown that for every ideal of , there exists such that . By Mcknight Theorem [7, Theorem 1.3], the set is . Dietrich Jr. has also proved that [7, Lemma 1.6] and if , then .

Theorem 4. Let be an infinite completely regular space. The following are equivalent: (1) is dense--separable; (2) is -cogenerated; (3) is -separable.

Proof. (1)(2): let . For each , there exists such that . By the previous observations from [7] we have but ; therefore , and hence by Lemma 1, is dense in . Since is a dense--separable space, there exists an -subset of , such that . Hence there exists an -set such that is dense in ; that is, . Now by Lemma 2, , and this latter observation in its turn shows that .
(2)(3) it is evident.
(3)(1): let be a dense subset of . Then by Lemma 1, . Now by Lemma 2, . Since is -separable, there is an -subset of such that , and again by Lemma 1, this shows that is dense in and hence dense in .

Observe that when is finite, is artinian and hence -cogenerated. If is separable, then is dense-separable. A ring is called von-Neumann regular if for every , there exists such that . I. Kaplansky has shown that every ideal in a commutative von-Neumann regular ring can be written as the intersection of some family of maximal ideals. Hence, a commutative von-Neumann regular ring is -separable if and only if it is -cogenerated. This implies that for any p-space , is -separable if and only if is -cogenerated. A ring is called right -ring (after Villamayor) if every right simple -module is injective. It is well known that over right -rings, every submodule of a right -module can be written as the intersection of a family of maximal submodules of . Hence over a right -ring, a right -module is -cogenerated if and only if it is -separable. Let be a family of ideals of ; we have . Hence as far as one is concerned with two sided ideals of , one obtains that is -separable (-cogenerated, resp.) when is -separable (-cogenerated, resp.).

Corollary 5. The following are equivalent: (1) is dense--separable; (2) is -cogenerated; (3) is -separable; (4) is -cogenerated; (5) is -separable; (6) is -cogenerated; (7) is -separable.

Proof. It is well known that , and by Theorem 4 the verification is immediate.

Corollary 6. If is either (1) separable metric or (2) separable and ordered, then is -cogenerated.

Proof. By [5, Corollary 3.2.], for these two cases is dense-separable. Now by Theorem 3 the proof is thorough.

When is separable, then is also separable. However, the converse is not true. In [5, example 5.3], a separable compact space has been introduced which is not dense-separable; for the space , is not separable. Otherwise should be dense-separable which is not the case. Based on these observations we have the following.

Example 7. There exists a separable space , such that is not separable.

However, the converse is true when we have a much stronger property as we observe in the next proposition.

Proposition 8. Let be a commutative ring. Then is -separable if and only if is dense--separable.

Proof. Let be dense--separable and a family of maximal ideals with zero intersection. This family then will be dense in . By dense--separability of , has an -subset with zero intersection. Let be an -separable ring and a dense subspace of . By definition has a zero intersection and by -separability of , it has an -subspace which is dense in .

According to Gelfand-Naimark's theorem every commutative -algebra with identity is isomorphic to , where is a suitable compact Hausdorff space. Based on Theorem 3 and Gelfand-Naimark's theorem, we have the following.

Corollary 9. Let be a commutative -algebra with identity and an arbitrary regular cardinal. Then the following are equivalent: (1) is -separable;(2) is -cogenerated.

Acknowledgment

The author would like to thank Mr. Olfati for his useful comments and discussion on the subject.