%0 Journal Article
%T The $\lambda$-super socle of the ring of continuous functions
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Mehran, Simin
%A Namdari, Mehrdad
%D 2017
%\ 01/01/2017
%V 6
%N Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
%P 37-50
%! The $\lambda$-super socle of the ring of continuous functions
%K $\lambda$-super socle
%K $\lambda$-isolated point
%K one point $\lambda$-compactification
%K $p_\lambda$-space
%R
%X The concept of $\lambda$-super socle of $C(X)$, denoted by $S_\lambda(X)$ (i.e., the set of elements of $C(X)$ such that the cardinality of their cozerosets are less than $\lambda$, where $\lambda$ is a regular cardinal number with $\lambda\leq |X|$) is introduced and studied. Using this concept we extend some of the basic results concerning $SC_F(X)$, the super socle of $C(X)$ to $S_\lambda(X)$, where $\lambda \geq\aleph_0$. In particular, we determine spaces $X$ for which $SC_F(X)$ and $S_\lambda(X)$ coincide. The one-point $\lambda$-compactification of a discrete space is algebraically characterized via the concept of $\lambda$-super socle. In fact we show that $X$ is the one-point $\lambda$-compactification of a discrete space $Y$ if and only if $S_\lambda(X)$ is a regular ideal and $S_\lambda(X)=O_x$, for some $x\in X$.
%U https://cgasa.sbu.ac.ir/article_33814_ae287573db032d67df112083dcb83c8f.pdf