TOWARDS A THEORY OF GORENSTEIN m-PRIMARY INTEGRALLY CLOSED IDEALS

Abstract. Let A be a Noetherian local ring with the maximal ideal m and d = dim A. The set XA of Gorenstein m-primary integrally closed ideals in A is explored in this paper. If k = A/m is algebraically closed and d ≥ 2, then XA 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) XA = {m}, and (3) k = A/m. When d = 1, XA is finite if and only if bA/p is not a DVR for any p ∈ Min bA, where bA denotes the m-adic completion. The list of elements in XA is given, when A is a one-dimensional Noetherian complete local integral domain. 1. Introduction. Let A be a Noetherian local ring with the maximal ideal m and d = dimA. Let I be an ideal in A and x ∈ A. Then we say that x is integral over I, if there exists a monic equation xn + c1xn−1 + · · ·+ cn =