Some new classes of topological spaces and annihilator ideals

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 F α -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 ) .