On -Cogenerated Commutative Unital -Algebras

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-？ech 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