On Auslander-Type Conditions of Modules

Abstract

For a left and right Noetherian ring $R$, we give some equivalent characterizations for ${}_{R}R$ satisfying the Auslander condition in terms of the flat (resp. injective) dimensions of the terms in a minimal injective coresolution (resp. flat resolution) of left $R$-modules. Furthermore, we prove that for an artin algebra $R$ satisfying the Auslander condition, $R$ is Gorenstein if and only if the subcategory consisting of finitely generated modules satisfying the Auslander condition is contravariantly finite. As applications, we get some equivalent characterizations of Auslander–Gorenstein rings and Auslander-regular rings.