Some new classes of topological spaces and annihilator ideals

Abstract

By a characterization of semiprime SA-rings by Birkenmeier, Ghirati and Taherifar in [4, Theorem 4.4], and by the topological characterization of C(X) as a Baer-ring by Stone and Nakano in [11, Theorem 3.25], it is easy to see that C(X) is an SA-ring (resp., IN-ring) if and only if X is an extremally disconnected space. This result motivates the following questions: Question (1): What is X if for any two ideals I and J of C(X) which are generated by two subsets of idempotents, Ann(I)+Ann(J)=Ann(I∩J)? Question (2): When does for any ideal I of C(X) exists a subset S of idempotents such that Ann(I)=Ann(S)? Along the line of answering these questions we introduce two classes of topological spaces. We call X an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). Topological properties of EF (resp., EZ)-spaces are investigated. As a consequence, a completely regular Hausdorff space X is an Fα-space in the sense of Comfort and Negrepontis for each infinite cardinal α if and only if X is an EF and EZ-space. Among other things, for a reduced ring R (resp., J(R)=0) we show that Spec(R) (resp., Max(R)) is an EZ-space if and only if for every ideal I of R there exists a subset S of idempotents of R such that Ann(I)=Ann(S).