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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.'
address: 'Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan'
author:
- Toshinori Kobayashi
title: 'Local rings with a self-dual maximal ideal'
---

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Introduction
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Let $R$ be a Cohen-Macaulay local ring with a canonical module $\omega$. For an $R$-module $M$, we denote by $M^\dag$ the $R$-module $\Hom_R(M,\omega)$. The $R$-module $M$ is called [*self-dual*]{} if there exists an isomorphism $M\xrightarrow[]{\cong} M^\dag$ of $R$-modules. Note that the self-duality of $R$-modules is independent of the choice of $\omega$.

Let $R$ and $S$ be artinian local rings such that $S$ maps onto $R$. Denote by $c_S(R)$ the colength $\ell_S(S)-\ell_S(R)$. In the 