Projectively full ideals in Noetherian rings, a survey

Abstract

AbstractWe discuss projective equivalence of ideals in Noetherian rings and the existenceor failure of existence of projectively full ideals. We describe connections with theRees valuations and Rees integers of an ideal, and consider the question of whetherimprovements can be made by passing to an integral extension ring. 1 Definitions, summary and examples. Let R be a Noetherian ring and let I 6= R be a regular ideal of R. (So I contains an elementwith zero annihilator.)Definition 1.1 An ideal J of R is projectively equivalent to I if there exist positiveintegers m and n such that I m and J n have the same integral closure, i.e., (I m ) a = (J n ) a .Projective equivalence is an equivalence relation on the regular proper ideals of R. LetP(I) denote the set of integrally closed ideals projectively equivalent to I.Remark 1.2 The set P(I) is discrete and linearly ordered with respect to inclusion. More-over, J and K in P(I) and m and n positive integers implies (J m K n ) a ∈ P(I). Thus thereis naturally associated to I and P(I) a unique subsemigroup S(I) of the additive semi-group of nonnegative integers IN