The Auslander-Reiten conjecture for Gorenstein rings

TL;DR

This paper proves that the Auslander-Reiten conjecture for Gorenstein rings in codimension one implies its validity for the entire ring, advancing understanding in ring theory.

Contribution

It establishes a reduction from codimension one cases to the full Gorenstein ring for the Auslander-Reiten conjecture.

Findings

Proves the conjecture in codimension one implies it for the whole ring.

Links the Auslander-Reiten conjecture to the Nakayama conjecture in Gorenstein rings.

Provides a new approach to verify the conjecture in broader cases.

Abstract

The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this note is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$.