Gorenstein FP∞-Injective Modules and w-Noetherian Rings

Abstract

We study some homological properties of Gorenstein [Formula: see text]-injective modules, and we prove (1) a ring [Formula: see text]is not necessarily coherent if every Gorenstein [Formula: see text]-injective [Formula: see text]-module is injective, and (2) a ring [Formula: see text] is not necessarily coherent if every Gorenstein injective [Formula: see text]-module is injective. In addition, we characterize [Formula: see text]-Noetherian rings in terms of Gorenstein [Formula: see text]-injective modules, and we prove that a ring [Formula: see text] is [Formula: see text]-Noetherian if and only if every GV-torsion-free FP-injective [Formula: see text]-module is Gorenstein [Formula: see text]-injective, if and only if any direct sum of GV-torsion-free FP-injective [Formula: see text]-modules is Gorenstein [Formula: see text]-injective.