On a -Kasch spaces

Ali Akbar Estaji; Melvin Henriksen

Archivum Mathematicum (2010)

  • Volume: 046, Issue: 4, page 251-262
  • ISSN: 0044-8753

Abstract

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If X is a Tychonoff space, C ( X ) its ring of real-valued continuous functions. In this paper, we study non-essential ideals in C ( X ) . Let a be a infinite cardinal, then X is called a -Kasch (resp. a ¯ -Kasch) space if given any ideal (resp. z -ideal) I with gen ( I ) < a then I is a non-essential ideal. We show that X is an ℵ 0 -Kasch space if and only if X is an almost P -space and X is an ℵ 1 -Kasch space if and only if X is a pseudocompact and almost P -space. Let C F ( X ) denote the socle of C ( X ) . For a topological space X with only a finite number of isolated points, we show that X is an a -Kasch space if and only if C ( X ) C F ( X ) is an a -Kasch ring.

How to cite

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Estaji, Ali Akbar, and Henriksen, Melvin. "On $a$-Kasch spaces." Archivum Mathematicum 046.4 (2010): 251-262. <http://eudml.org/doc/116490>.

@article{Estaji2010,
abstract = {If $X$ is a Tychonoff space, $C(X)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C(X)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\bar\{a\}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $\operatorname\{gen\}\,(I)<a$ then $I$ is a non-essential ideal. We show that $X$ is an $\aleph _0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an $\aleph _1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F(X)$ denote the socle of $C(X)$. For a topological space $X$ with only a finite number of isolated points, we show that $X$ is an $a$-Kasch space if and only if $\frac\{C(X)\}\{C_F(X)\}$ is an $a$-Kasch ring.},
author = {Estaji, Ali Akbar, Henriksen, Melvin},
journal = {Archivum Mathematicum},
keywords = {$a$-Kasch space; almost $P$-space; basically disconnected; $C$-embedded; essential ideal; extremally disconnected; fixed ideal; free ideal; Kasch ring; $P$-space; pseudocompact space; Stone-Čech compactification; socle; realcompactification; -Kasch space; almost -space; -embedded; essential ideal; extremally disconnected; fixed ideal; free ideal; Kasch ring; -space},
language = {eng},
number = {4},
pages = {251-262},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On $a$-Kasch spaces},
url = {http://eudml.org/doc/116490},
volume = {046},
year = {2010},
}

TY - JOUR
AU - Estaji, Ali Akbar
AU - Henriksen, Melvin
TI - On $a$-Kasch spaces
JO - Archivum Mathematicum
PY - 2010
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 046
IS - 4
SP - 251
EP - 262
AB - If $X$ is a Tychonoff space, $C(X)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C(X)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\bar{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $\operatorname{gen}\,(I)<a$ then $I$ is a non-essential ideal. We show that $X$ is an $\aleph _0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an $\aleph _1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F(X)$ denote the socle of $C(X)$. For a topological space $X$ with only a finite number of isolated points, we show that $X$ is an $a$-Kasch space if and only if $\frac{C(X)}{C_F(X)}$ is an $a$-Kasch ring.
LA - eng
KW - $a$-Kasch space; almost $P$-space; basically disconnected; $C$-embedded; essential ideal; extremally disconnected; fixed ideal; free ideal; Kasch ring; $P$-space; pseudocompact space; Stone-Čech compactification; socle; realcompactification; -Kasch space; almost -space; -embedded; essential ideal; extremally disconnected; fixed ideal; free ideal; Kasch ring; -space
UR - http://eudml.org/doc/116490
ER -

References

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