Projectively equivalent ideals and Rees valuations

[摘要] Let R be a Noetherian ring. Two ideals I and J in R are projectively equivalent in case the integral closure of I-i is equal to the integral closure of J(j) for some i, j is an element of N+. It is known that if I and J are projectively equivalent, then the set Rees I of Rees valuation rings of I is equal to the set Rees J of Rees valuation rings of J and the values of I and J with respect to these Rees valuation rings are proportional. We observe that the converse also holds. In particular, if the ideal I has only one Rees valuation ring V, then the ideals J projectively equivalent to I are precisely the ideals J such that Rees J = {V}. In certain cases such as: (i) dim R = 1, or (ii) R is a two-dimensional regular local domain, we observe that if I has more than one Rees valuation ring, then there exist ideals J such that Rees I = Rees J, but J is not projectively equivalent to L If I and J are regular ideals of R, we prove that Rees I U Rees J subset of or equal to Rees I J with equality holding if dim R less than or equal to 2, but not holding in general if dim R greater than or equal to 3. We associate to I and to the set P(I) of integrally closed ideals projectively equivalent to I a numerical semigroup S(I) subset of or equal to N such that S(I) = N if and only if there exists J is an element of P(I) for which P(I) = {(J(n))(a) \n is an element of N+\. (C) 2004 Published by Elsevier Inc.

[发布日期] 2004-12-01