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

Authors : Somayeh SOLTANPOUR

Pages : 1430-1436

View : 21 | Download : 6

Publication Date : 2019-10-08

Article Type : Research Paper

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