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

F. Azarpanah ¹   ; O. A. S. Karamzadeh ¹   ; S. Rahmati ¹

¹ Department of Mathematics Chamran University Ahvaz, Iran

Colloquium Mathematicum, Tome 111 (2008) no. 2, pp. 315-336

Résumé

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.

Détail

DOI : 10.4064/cm111-2-9

Mots-clés : socle shown each prime ideal essential each prove every prime ideal resp z ideal essential only set zeros contains isolated points resp mathop int nolimits emptyset proved dim geq dim where dim denotes goldie dimension inequality may strict algebraic characterization compact spaces countable number nonisolated points each essential ideal observe essential only set isolated points finite finally characterize topological spaces which jacobson radical zero consequence observe cardinality discrete space nonmeasurable only upsilon realcompactification nbsp first countable