Local rings with self-dual maximal ideal

Abstract

Let $R$ be a Cohen–Macaulay local ring possessing a canonical module. In this paper, we consider when the maximal ideal of $R$ is self-dual—i.e., it is isomorphic to its canonical dual as an $R$-module. Local rings satisfying this condition are called Teter rings, studied by Teter, Huneke–Vraciu, Ananthnarayan–Avramov–Moore, and others. In the one-dimensional case, we show such rings are exactly the endomorphism rings of the maximal ideals of some Gorenstein local rings of dimension one. We also provide some connection between the self-duality of the maximal ideal and near Gorensteinness.