Communications of the
Korean Mathematical Society CKMS

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 $Uin Cozleft( Xight) $, and each $gin C^{ast }left( Uight) $ there is $Vin Cozleft( Xight) $ such that $Usubseteq V$, $overline{V}=X$, 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.