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\(P\)-spaces and Artin-Rees property. (Persian. English summary) Zbl 1304.54045

Summary: 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.

MSC:

54C40 Algebraic properties of function spaces in general topology
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
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