# Local rings with self-dual maximal ideal

## TL;DR

This paper investigates Cohen-Macaulay local rings with a canonical module where the maximal ideal is self-dual, linking such rings to endomorphism rings of maximal ideals in Gorenstein rings and exploring their near Gorenstein properties.

## Contribution

It characterizes when the maximal ideal of a Cohen-Macaulay local ring is self-dual, connecting it to endomorphism rings of Gorenstein rings and near Gorensteinness.

## Key findings

- Self-dual maximal ideals correspond to endomorphism rings of Gorenstein rings.
- Such rings are characterized in the positive dimension case.
- Connections between self-duality and near Gorensteinness are established.

## 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, and studied by Teter, Huneke-Vraciu, Ananthnarayan-Avramov-Moore, and so on. On the positive 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.