Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Projectively full ideals in Noetherian rings by William HEINZER, Purdue University Abstract: This is joint work with Catalin Ciuperca, Jack Ratliff and Dave Rush. Let I be a regular proper ideal of a Noetherian ring R. The set P(I) of integrally closed ideals projectively equivalent to I is linearly ordered with respect to inclusion, and there is naturally associated to I and P(I) a numerical semigroup S(I). The semigroup S(I) is the semigroup of nonnegative integers 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. We prove that if R contains the field of rational numbers, then there exists a finite free integral extension ring A of R such that P(IA) is projectively full; and if R is an integral domain, then there also exists a finite integral extension domain B of R such that P(IB) is projectively full.