Some new classes of topological spaces and annihilator ideals

TL;DR

This paper introduces new classes of topological spaces, EF and EZ spaces, to characterize properties of annihilator ideals in rings of continuous functions, linking topological and algebraic structures.

Contribution

It defines EF and EZ spaces, explores their properties, and connects them to the structure of annihilator ideals in rings like C(X) and reduced rings.

Findings

EF and EZ spaces are characterized by specific separation and closure properties.

A space being EF and EZ is equivalent to certain algebraic conditions in rings of continuous functions.

Spec(R) and Max(R) are EZ-spaces under conditions on 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 $Ann(I)=Ann(S)$.