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Several results are proved concerning the set A*(I) = {P E Spec R; P is a prime divisor of the integral closure (Γ) a of /' for all large /}, where / is an ideal in a Noetherian ring R. Among these are: if P is a prime divisor of (Γ) a for some i > 1, then P is a prime divisor of (I") a for all n > i\ a characterization of Cohen-Macaulay rings and of altitude two local UFDs in terms of A*(I); and, some results on the relationship of A*(I) to A*(IS) with S a flat Λ-algebra and to A*((I + z)/z) with z a minimal prime ideal in R.

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 ∈ 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 I . If I and J are regular ideals of R, we prove that Rees I ∪ Rees J ⊆ Rees I J with equality holding if dim R 2, but not holding in general if dim R 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 ) = {(J n ) a | n ∈ N + }.

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