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

• Autorzy: F. Azarpanah , O. A. S. Karamzadeh , S. Rahmati
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 16P20: Artinian rings and modules
• 54C40: Algebraic properties of function spaces
• 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2008
• Tom: 111
• Numer: 2
• Strony: 315-336

Daty

wydano

2008

Twórcy

autor

F. Azarpanah

• Department of Mathematics, Chamran University, Ahvaz, Iran

autor

O. A. S. Karamzadeh

• Department of Mathematics, Chamran University, Ahvaz, Iran

autor

S. Rahmati

• Department of Mathematics, Chamran University, Ahvaz, Iran

Identyfikatory

DOI

10.4064/cm111-2-9