On the locally socle of C(X) whose local cozeroset is cocountable (cofinite)

Abstract

Let $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). We introduce and study the concept of colocally socle of $C(X)$ as $C_{\mu}{S_{\lambda}}(X)=\left\{ f\in C(X):|X\backslash {S}^{\lambda}_{f}|<\mu \right\}$, where ${{S}^{\lambda}_{f}}$ is the union of all open subsets $U$ in $X$ such that $|U\backslash Z(f)|<\lambda$. $C_{\mu}{S_{\lambda}}(X)$ is a $z$-ideal of $C(X)$ containing ${{C}_{F}}(X)$. In particular, $C_{{\aleph}_0}{S_{{\aleph}_0}}(X)=CC_F(X)$ and $C_{{\aleph}_1}{S_{{\aleph}_1}}(X)=CS_c(X)$ are investigated. For each of the containments in the chain ${{C}_{F}}(X)\subseteq CC_F(X)\subseteq C_{\mu}{S_{\lambda}}(X)\subseteq C(X)$, we characterize the spaces $X$ for which the containment is actually an equality. We determine the conditions such that $CC_F(X)$ ($CS_c(X)$) is not prime in any subrings of $C(X)$ which contains the idempotents of $C(X)$. The primeness of $CC_F(X)$ in some subrings of $C(X)$ is investigated.