Proper Resolutions and Auslander-Type Conditions of Modules

We obtain some methods to construct a (strongly) proper resolution (resp. coproper coresolution) of one end term in a short exact sequence from that of the other two terms. By using this method, we prove that for a left and right Noetherian ring R, RR satisfies the Auslander condition if and only if so does every flat left R-module, if and only if the injective dimension of the ith term in a minimal flat resolution of any injective left R-module is at most i− 1 for any i ≥ 1, if and only if the flat (resp. injective) dimension of the ith term in a minimal injective (resp. flat) resolution of any left R-module M is at most the flat (resp. injective) dimension of M plus i − 1 for any i ≥ 1, if and only if the flat (resp. injective) dimension of the injective envelope (resp. flat cover) of any left R-module M is at most the flat (resp. injective) dimension of M , and if and only if any of the opposite versions of the above conditions hold true. Furthermore, we prove that for an Artinian 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.