On relative counterpart of Auslander’s conditions

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module [Formula: see text], we introduce weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are [Formula: see text]-Frobenius categories in characterizing weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings.

SG_C-Projective, Injective and Flat modules

In this paper we introduce the concepts of $SG_C$-projective, injective and flat modules, where $C$ is a semidualizing module and we discuss some connections among $SG_C$-projective, injective and flat modules.

Finitely projective modules with respect to a semidualizing module

Let [Formula: see text] be a commutative ring. We define and study [Formula: see text]-projective modules with respect to a semidualizing [Formula: see text]-module [Formula: see text], which are called [Formula: see text]–[Formula: see text]-projective modules. As consequences, we characterize several rings such as [Formula: see text]-coherent rings and Artinian rings using [Formula: see text]–[Formula: see text]-projective modules. Some known results are extended.