Resumen de When is the super socle of C(X) prime?

Sahar Ghasemzadeh, Mehrdad Namdari

• Let $SC_F(X)$ denote the ideal of $C(X)$ consisting of functions which are zero everywhere except on a countable number of points of $X$. It is generalization of the socle of C(X) denoted by $C_F(X)$. Using this concept we extend some of the basic results concerning $C_F(X)$ to $SC_F(X)$. In particular, we characterize the spaces $X$ such that $SC_F(X)$ is a prime ideal in $C(X)$ (note, $C_F(X)$ is never prime ideal in $C(X)$). This may be considered as an advantage of $SC_F(X)$ over $C_F(X)$. We are also interested in characterizing topological spaces $X$ such that $C_c(X)=\mathbb{R}+SC_F(X)$, where $C_c(X)$ denotes the subring of $C(X)$ consisting of functions with countable image.