Shahid Chamran University of AhvazJournal of Advanced Mathematical Modeling2251-80884220151023Locally Socle of C(X)Locally Socle of C(X)879911215FASomayehSoltanpourDepartment of Mathematics, Shahid Chamran universityMehrdadNamdariDepartment of Mathematics,
Shahid Chamran University0000-0003-0966-7234Journal Article20140903Let $LC_F(X)$ be the socle of $C(X)$ and $LC_F(X)={fin C(X) : overline{S_f}=X}$ , where $S_f$ is the union of all open subsets $U$ in $X$ such that $Ubackslash Z(f)|<infinity|$, $LC_F(X)$ is called the locally socle of $C(X)$ and it is a $z$-ideal of $C(X)$ containing $C_F(X)$. We characterize spaces $X$ for which the equality in the relation $C_F(X)subseteq LC_F(X)subseteq C(X)$ is hold. In fact, we show that $X$ is an almost discrete space if and only if $LC_F(X)=C(X)$. We note that if $X$ is an infinite space, then $C_F(X)subsetneq C(X)$. We also observe that $|I(X)|<infty$ if and only if $LC_F(X)=C_F(X)$. Moreover, it is shown that if $|I(X)|<infty$, then $LC_F(X)$ is never essential in any subring of $C(X)$ , while $LC_F(X)$ is an intersection of essential ideals of $C(X)$. We determine the conditions such that $LC_F(X)$ is not prime in any subring of $C(X)$ which contains the idempotents of $C(X)$. We investigate the primness of $LC_F(X)$ in some subrings of $C(X)$ .Let $LC_F(X)$ be the socle of $C(X)$ and $LC_F(X)={fin C(X) : overline{S_f}=X}$ , where $S_f$ is the union of all open subsets $U$ in $X$ such that $Ubackslash Z(f)|<infinity|$, $LC_F(X)$ is called the locally socle of $C(X)$ and it is a $z$-ideal of $C(X)$ containing $C_F(X)$. We characterize spaces $X$ for which the equality in the relation $C_F(X)subseteq LC_F(X)subseteq C(X)$ is hold. In fact, we show that $X$ is an almost discrete space if and only if $LC_F(X)=C(X)$. We note that if $X$ is an infinite space, then $C_F(X)subsetneq C(X)$. We also observe that $|I(X)|<infty$ if and only if $LC_F(X)=C_F(X)$. Moreover, it is shown that if $|I(X)|<infty$, then $LC_F(X)$ is never essential in any subring of $C(X)$ , while $LC_F(X)$ is an intersection of essential ideals of $C(X)$. We determine the conditions such that $LC_F(X)$ is not prime in any subring of $C(X)$ which contains the idempotents of $C(X)$. We investigate the primness of $LC_F(X)$ in some subrings of $C(X)$ .https://jamm.scu.ac.ir/article_11215_9556bc6c71c251bc71d7a0abb6004cdd.pdf