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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; m) be d-Gorenstein, d â¥ 2, and let Î be a Noetherian
R-algebra which is Gorenstein and (maximal) Cohen-Macaulay as R-module. If M is
an Artinian self-orthogonal Gorenstein injective Î-module such that HomÎ(Îp; M) is
an injective Îp-module for every nonmaximal prime ideal p of R, then M is injective.
Some applications are discussed afterwards.
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