Abstract
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.
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