When is $C(X)$ an EM-ring?
Commun. Korean Math. Soc. 2022 Vol. 37, No. 1, 17-29
Published online July 7, 2021
Printed January 31, 2022
Emad Abuosba, Isaaf Atassi
The University of Jordan; The University of Jordan
Abstract : A commutative ring with unity $R$ is called an EM-ring if for any finitely generated ideal $I$ there exist $a$ in $R$ and a finitely generated ideal $J $ with $\limfunc{Ann}(J)=0$ and $I=aJ$. In this article it is proved that $ C(X)$ is an EM-ring if and only if for each $U\in Coz\left( X\right) $, and each $g\in C^{\ast }\left( U\right) $ there is $V\in Coz\left( X\right) $ such that $U\subseteq V$, $\overline{V}=X$, and $g$ is continuously extendable on $V$. Such a space is called an EM-space. It is shown that EM-spaces include a large class of spaces as F-spaces and cozero complemented spaces.\ It is proved among other results that $X$ is an EM-space if and only if the Stone-\v{C}ech compactification of $X$ is.
Keywords : $C(X)$, EM-ring, generalized morphic ring, PP-ring, PF-ring, basically disconnected space, F-space, cozero complemented space
MSC numbers : 13A15, 54C30, 54C45
Supported by : This article is a part of the Ph.D. thesis prepared by the second author under supervision of the first in The University of Jordan.
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