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 + }.

Let R be a Noetherian commutative ring with unit 1 = 0, and let I be a regular proper ideal of R. The set P(I ) of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and discrete. There is naturally associated to P(I ) a numerical semigroup S(I ); we have S(I ) = N if and only if every element of P(I ) is the integral closure of a power of the largest element J of P(I ). If this holds, the ideal J and the set P(I ) are said to be projectively full. If I is invertible and R is integrally closed, we prove that P(I ) is projectively full. We investigate the behavior of projectively full ideals in various types of ring extensions. We prove that a normal ideal I of a local ring (R, M) is projectively full if I M 2 and both the associated graded ring G(M) and the fiber cone ring F (I ) are reduced. We present examples of normal local domains (R, M) of altitude two for which the maximal ideal M is not projectively full.

Let A be a local ring with maximal ideal m. For an arbitrary ideal I of A, we define the generalized Hilbert coefficientsWhen the ideal I is m-primary, j k (I) = (0, . . . , 0, (−1) k e k (I)), where e k (I) is the classical k th Hilbert coefficient of I. Using these coefficients we give a numerical characterization of the homogeneous components of the S 2 -ification of S = A[It,t −1 ], extending to not necessarily m-primary ideals the results obtained in [7].

In this paper, we introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf rings, and the strict closedness of rings. Furthermore, we give various examples of weakly Arf rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.

Let R be a Noetherian commutative ring with unit 1 = 0, and let I be a regular proper ideal of R. The set P(I ) of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and discrete. There is naturally associated to I and to P(I ) a numerical semigroup S(I ); we have S(I ) = N if and only if every element of P(I ) is the integral closure of a power of the largest element K of P(I ). If this holds, the ideal K and the set P(I ) are said to be projectively full. A special case of the main result in this paper shows that if R contains the rational number field Q, then there exists a finite free integral extension ring A of R such that P(I A) is projectively full. If R is an integral domain, then the integral extension A has the property that P((I A + z * )/z * ) is projectively full for all minimal prime ideals z * in A. Therefore in the case where R is an integral domain there exists a finite integral extension domain B = A/z * of R such that P(I B) is projectively full.