Auslander-Reiten Conjecture in a Dual Vein

Abstract

We consider a dual notion of the famous Auslander-Reiten Conjecture in case of Noetherian algebras over commutative Noetherian rings. Firstly, in the introduction, we will examine its relevance by showing that in an standard situation, the validity of this dual implies the validity of the Auslander-Reiten Conjecture itself. Moreover, in two important cases these two notions coincide: Artin algebras, and Noetherian algebras over complete local Noetherian rings. In this regard we will prove the following theorem: Let $(R, \mathfrak {m})$ be d-Gorenstein, d ≥ 2, and let Λ be a Noetherian R-algebra which is Gorenstein and (maximal) Cohen-Macaulay as R-module. If M is an Artinian self-orthogonal Gorenstein injective Λ-module such that ${\text {Hom}}_{\Lambda }({\Lambda }_{\mathfrak {p}}, M)$ is an injective ${\Lambda }_{\mathfrak {p}}$-module for every nonmaximal prime ideal $\mathfrak {p}$ of R, then M is injective. Some applications are discussed afterwards.