Projectively full ideals in Noetherian rings (II)

AbstractLet 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(IA) is projectively full. If R is an integral domain, then the integral extension A has the property that P((IA+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(IB) is projectively full