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Paper IPM / M / 16378  


Abstract:  
We consider a dual notion of the famous AuslanderReiten 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 AuslanderReiten 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 dGorenstein, d â¥ 2, and let Î be a Noetherian
Ralgebra which is Gorenstein and (maximal) CohenMacaulay as Rmodule. If M is
an Artinian selforthogonal Gorenstein injective Îmodule such that HomÎ(Îp; M) is
an injective Îpmodule for every nonmaximal prime ideal p of R, then M is injective.
Some applications are discussed afterwards.
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