# On the vanishing of self extensions over Cohen-Macaulay local rings

## TL;DR

This paper extends known results about the vanishing of self extensions of modules, originally established over Gorenstein rings, to Cohen-Macaulay local rings with canonical modules, advancing the understanding of the Auslander-Reiten Conjecture.

## Contribution

It generalizes existing theorems on self extension vanishing from Gorenstein rings to Cohen-Macaulay local rings with canonical modules.

## Key findings

- Recovered theorems of Araya, Ono, and Yoshino.
- Extended the scope of the Auslander-Reiten Conjecture.
- Provided new conditions under which self extensions vanish.

## Abstract

The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the long-standing conjectures in ring theory. Although it is still open, there are several results in the literature that establish the conjecture over Gorenstein rings under certain conditions. The purpose of this article is to obtain extensions of such results over Cohen-Macaulay local rings that admit canonical modules. In particular, our main result recovers theorems of Araya, and Ono and Yoshino simultaneously.