Projectively equivalent ideals and Rees valuations

Let R be a Noetherian ring. Two ideals I and J in R are projectively equivalent in case the integral closure of Ii is equal to the integral closure of Jj for some i,j N+. It is known that if I and J are projectively equivalent, then the set ReesI of Rees valuation rings of I is equal to the set ReesJ 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 ReesJ= V . In certain cases such as: (i) dimR=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 ReesI=ReesJ, but J is not projectively equivalent to I. If I and J are regular ideals of R, we prove that ReesI ReesJ ReesIJ with equality holding if dimR 2, but not holding in general if dimR 3. We associate to I and to the set P(I) of integrally closed ideals projectively equivalent to I a numerical semigroup S(I) N such that S(I)=N if and only if there exists J P(I) for which P(I)= (Jn)an N+ .