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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 rings with a unique proper essential right ideal

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Karamzadeh O., Motamedi M., Shahrtash S.
Fundamenta Mathematicae
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2004
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tom 183
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nr 3
229-244
EN
Right ue-rings (rings with the property of the title, i.e., with the maximality of the right socle) are investigated. It is shown that a semiprime ring R is a right ue-ring if and only if R is a regular V-ring with the socle being a maximal right ideal, and if and only if the intrinsic topology of R is non-discrete Hausdorff and dense proper right ideals are semisimple. It is proved that if R is a right self-injective right ue-ring (local right ue-ring), then R is never semiprime and is Artin semisimple modulo its Jacobson radical (R has a unique non-zero left ideal). We observe that modules with Krull dimension over right ue-rings are both Artinian and Noetherian. Every local right ue-ring contains a duo subring which is again a local ue-ring. Some basic properties of right ue-rings and several important examples of these rings are given. Finally, it is observed that rings such as C(X), semiprime right Goldie rings, and some other well known rings are never ue-rings.
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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
4
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Erratum to "On rings with a unique proper essential right ideal" (Fund. Math. 183 (2004), 229-244)

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Karamzadeh O., Motamedi M., Shahrtash S.
Fundamenta Mathematicae
|
2009
|
tom 205
|
nr 3
289-291
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