Some new classes of topological spaces and annihilator ideals

Abstract: By a characterization of semiprime $SA$-rings by Birkenmeier, Ghirati and Taherifar in \cite[Theorem 4.4]{B}, and by the topological characterization of $C(X)$ as a Baer-ring by Stone and Nakano in \cite[Theorem 3.25]{KM}, it is easy to see that $C(X)$ is an $SA$-ring (resp., $IN$-ring) \ifif $X$ 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, $Ann(I)+Ann(J)=Ann(I\cap J)?$ 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 $\textit{EF}$ (resp., $\textit{EZ}$)-$\textit{space}$ 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 $\textit{EF}$ (resp., $\textit{EZ}$)-$\textit{spaces}$ are investigated. As a consequence, a completely regular Hausdorff space $X$ is an $F_{\alpha}$-space in the sense of Comfort and Negrepontis for each infinite cardinal $\alpha$ \ifif $X$ 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 \ifif for every ideal $I$ of $R$ there exists a subset $S$ of idempotents of $R$ such that $Ann(I)=Ann(S)$.