Nagata rings

Abstract

Let A be a commutative ring. For any set p of prime ideals of A, we define a new ring Na(A, p): the Nagata ring. This new ring has the particularity that we may transform certain properties relative to p to properties on the whole ring Na(A, p); some of these properties are: ascending chain condition, Krull dimension, Cohen-Macaulay, Gorenstein. Our main aim is to show that most of the above properties relative to a set of prime ideals p(i.e., local properties) determine and are determined by the same properties on the Nagata ring (i.e., global properties). In order to look for new applications, we show that this construction is functorial, and exhibits a functorial embedding from the localized category (A, p)-Mod into the module category Na(A, p)-Mod.

Keywords

Cohen–Macaulay / Gorenstein / Krull / Nagata rings

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Higher Education Press and Springer-Verlag Berlin Heidelberg