## $C(X)$ vs. $C(X)$ modulo its socle

### Volume 111 / 2008

#### Abstract

Let $C_F(X)$ be the socle of $C(X)$. It is shown that each prime ideal in ${C(X)}/{C_F(X)}$ is essential. For each $h\in C(X)$, we prove that every prime ideal (resp. z-ideal) of ${C(X)}/{(h)}$ is essential if and only if the set $Z(h)$ of zeros of $h$ contains no isolated points (resp. $\mathop{\rm int}\nolimits Z(h)=\emptyset$). It is proved that $\dim ({C(X)}/{C_F(X)}) \geq \dim C(X)$, where $\dim C(X)$ denotes the Goldie dimension of $C(X)$, and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential ideal $E$ in $C(X)$, we observe that ${E}/{C_F(X)}$ is essential in ${C(X)}/{C_F(X)}$ if and only if the set of isolated points of $X$ is finite. Finally, we characterize topological spaces $X$ for which the Jacobson radical of ${C(X)}/{C_F(X)}$ is zero, and as a consequence we observe that the cardinality of a discrete space $X$ is nonmeasurable if and only if $\upsilon X$, the realcompactification of $X$, is first countable.