P-spaces and Artin-Rees Property

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.