P-spaces and Artin-Rees Property

Authors

Department of Mathematics, Shahid Chamran University, Ahvaz, Iran

Abstract

In this article, we study the Artin-Rees property in  C(X), in the  rings of fractions of  C(X) and in the factor rings of C(X) . We show that C(X)/(f) is an Artin-Rees ring if and only if  Z(f)  is an open P-space. A necessary and sufficient condition for the local rings of  C(X)   to be Artin-Rees rings is that each prime ideal in  C(X)  becomes minimal and it turns out that every local ring of C(X)  is an Artin-Rees ring if and only if  X  is a P-space. Finally we have shown that whenever XZ(f)  is dense  C-embedded in  X , then  C(X)f  is regular if and only if  Xz(f) is a P-space.

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