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## Colloquium Mathematicum

2008 | 111 | 2 | 315-336
Tytuł artykułu

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

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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 ∈ 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. int Z(h) = ∅). It is proved that $dim (C(X)/C_{F}(X)) ≥ dim C(X)$, 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 υX, the realcompactification of X, is first countable.
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315-336
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wydano
2008
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autor
• Department of Mathematics, Chamran University, Ahvaz, Iran
autor
• Department of Mathematics, Chamran University, Ahvaz, Iran
autor
• Department of Mathematics, Chamran University, Ahvaz, Iran
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