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

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Azarpanah F., Karamzadeh O., Rahmati S.
Colloquium Mathematicum
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2008
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tom 111
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nr 2
315-336
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.
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On z◦ -ideals in C(X)

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Azarpanah F., O. A. S Karamzadeh, Rezai Aliabad A.
Fundamenta Mathematicae
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1999
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tom 160
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nr 1
15-25
EN
An ideal I in a commutative ring R is called a z°-ideal if I consists of zero divisors and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We characterize topological spaces X for which z-ideals and z°-ideals coincide in , or equivalently, the sum of any two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and P-spaces are characterized in terms of z°-ideals. Finally, we construct two topological almost P-spaces X and Y which are not P-spaces and such that in every prime z°-ideal is either a minimal prime ideal or a maximal ideal and in C(Y) there exists a prime z°-ideal which is neither a minimal prime ideal nor a maximal ideal.
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Erratum to: "C(X) vs. C(X) modulo its socle" (Colloq. Math. 111 (2008), 315-336)

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Azarpanah F., Karamzadeh O., Rahmati S.
Colloquium Mathematicum
|
2015
|
tom 139
|
nr 1
147-148
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