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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 67, Issue 4


The super socle of the ring of continuous functions

Sahar Ghasemzadeh / Omid A. S. Karamzadeh / Mehrdad Namdari
Published Online: 2017-07-14 | DOI: https://doi.org/10.1515/ms-2017-0028


We introduce and study the concept of the super socle of C(X), denoted by SCF(X) (i.e., the set of elements of C(X), which are zero everywhere except on a countable number of points of X). Using this concept we extend some of the basic results concerning CF(X), the socle of C(X), to SCF(X). In particular, we determine spaces X such that CF(X) and SCF(X) coincide. Spaces X such that Ann(SCF(X)) is generated by an idempotent are fully characterized. It is shown that SCF(X) is an essential ideal in C(X) if and only if the set of countably isolated points (i.e., points with countable neighborhoods) of X is dense in X. The one-point Lindelöffication of uncountable discrete spaces is algebraically characterized via the concept of the super socle. Consequently, it is observed that whenever OxSCF(X) and SCF(X) is a regular ideal (von Neumann), then X is either a countable discrete space or the one-point Lindelöffication of an uncountable discrete space. Consequently, in this case SCF(X) is a prime ideal in C(X) (note, CF(X) is never prime C(X))

MSC 2010: Primary 54C30; 54C40; 54G12; Secondary 13C11; 16H20

Keywords: super socle of C(X); countably isolated point; countably discrete space; c-disjoint spaces; pseudo minimal ideals; one-point Lindelöffication


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About the article

Received: 2015-05-12

Accepted: 2015-07-01

Published Online: 2017-07-14

Published in Print: 2017-08-28

Citation Information: Mathematica Slovaca, Volume 67, Issue 4, Pages 1001–1010, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0028.

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