Projectively full ideals in Noetherian rings (II)

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 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.