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Volume 2013 (2013), Article ID 452862, 3 pages
On -Cogenerated Commutative Unital -Algebras
Department of Mathematics, Yasouj University, Yasouj, Iran
Received 22 November 2012; Accepted 17 May 2013
Academic Editor: Xueqing Chen
Copyright © 2013 Ehsan Momtahan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  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 , 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 . 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 , 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 , 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 , 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 ). 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 . 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 , page 195, example 3). Since every dense subset of is also dense in , the Sorgenfrey line is dense-separable. In , 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  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  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.
The author would like to thank Mr. Olfati for his useful comments and discussion on the subject.
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