几类新的拓扑空间与零化子理想
Some new classes of topological spaces and annihilator ideals
摘要：
通过Birkenmeier、Ghirati和Taherifar在\cite [定理4.4]{B}中对半素$SA$ -环的刻画，以及Stone和Nakano在\cite [定理3.25]{KM}中对$C(X)$作为Baer-环的拓扑刻画，很容易看出$C(X)$是$SA$ -环（分别为，$IN$ -环）\ifif $X$是一个极不连通空间。这一结果激发了以下问题：问题$(1)$：什么是PDMATHAMR，如果任何两个理想PDMATHAMT和$J$的$C(X)$这是由两个子集的幂等元，$Ann(I)+Ann(J)=Ann(I\cap J)?$问题$(2)$：当任何理想PDMATHAMT的$C(X)$存在一个子集$S$的幂等元，使$Ann(I)=Ann(S)$？沿着回答这些问题的路线，我们引入两类拓扑空间。我们将$X$称为$\textit{EF}$（分别，$\textit{EZ}$）- $\textit{space}$，如果闭集的不相交并是完全分离的（分别地，每个正则闭子集是闭子集的并集的闭包）。$\textit{EF}$的拓扑性质（分别，$\textit{EZ}$）- $\textit{spaces}$。因此，一个完全正则的Hausdorff空间$X$是一个舒适和Negrepontis意义下的$F_α$ -空间，对于每个无限基数$α$ \ifif $X$是一个$EF$和$EZ$ -空间。除其他外，对于减少环$R$（分别，$J(R)=0$），我们表明，$Spec(R)$（分别，$Max(R)$）是$EZ$ -空间\ifif对于$R$的每个理想$I$，存在$R$的幂等元的子集$S$，使得$Ann(I)=Ann(S)$ .
英文摘要：
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 $F_α$-space in the sense of Comfort and Negrepontis for each infinite cardinal $α$ \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 $Ann(I)=Ann(S)$.
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