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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Intersection of essential ideals in $C(X)$

Author: F. Azarpanah
Journal: Proc. Amer. Math. Soc. 125 (1997), 2149-2154
MSC (1991): Primary 54C40
MathSciNet review: 1422843
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Abstract: The infinite intersection of essential ideals in any ring may not be an essential ideal, this intersection may even be zero. By the topological characterization of the socle by Karamzadeh and Rostami (Proc. Amer. Math.Soc. 93 (1985), 179-184), and the topological characterization of essential ideals in Proposition 2.1, it is easy to see that every intersection of essential ideals of $C(X)$ is an essential ideal if and only if the set of isolated points of $X$ is dense in $X$. Motivated by this result in $C(X)$, we study the essentiallity of the intersection of essential ideals for topological spaces which may have no isolated points. In particular, some important ideals $C_K(X)$ and $C_\infty (X)$, which are the intersection of essential ideals, are studied further and their essentiallity is characterized. Finally a question raised by Karamzadeh and Rostami, namely when the socle of $C(X)$ and the ideal of $C_K(X)$ coincide, is answered.

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F. Azarpanah
Affiliation: Department of Mathematics, The University, Ahvaz, Iran

Keywords: Essential ideals, socle, almost locally compact, pseudo-discrete, first category, nowhere dense
Received by editor(s): January 20, 1995
Communicated by: James West
Article copyright: © Copyright 1997 American Mathematical Society