Projectively full ideals in Noetherian rings

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⊈M2 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.