Towards a Theory of Gorenstein

Abstract

Let A be a Noetherian local ring with the maximal ideal m and d = dim A. The set Χ A of Gorenstein m-primary integrally closed ideals in A is explored in this paper. If k = A/m is alge- braically closed and d≥2, then ΧA is infinite. In contrast, for each field k which is not algebraically closed and for each integer d ≥ 0, there exists a Noetherian complete equi-characteristic local integral domain A with dim A = d such that (1) the normalization of A is regular, (2) ΧA = {m}, and (3) k=A/m. When d = 1, ΧA is finite if and only if A/p is not a DVR for any p E Min A, where A denotes the m-adic completion. The list of elements in ΧA is given, when A is a one-dimensional Noetherian complete local integral domain.