Duality Pairs Induced by Gorenstein Projective Modules with Respect to Semidualizing Modules

Abstract

Let C be a semidualizing module over a commutative Noetherian ring R. We investigate duality pairs induced by C-Gorenstein projective modules. It is proven that R is Artinian if and only if (GP C , GI C ) is a duality pair if and only if (GI C , GP C ) is a duality pair and M + ∈ GI C whenever M ∈ GP C , where GP C (GI C ) is the class of C-Gorenstein projective (C-Gorenstein injective) R-modules. In particular, we give a necessary and sufficient condition for a commutative Artinian ring to be virtually Gorenstein. Moreover, we get that R is Artinian if and only if the class GP of Gorentein projective R-modules is preenveloping. As applications, some new criteria for a semidualizing module to be dualizing are given provided that R is a commutative Artinian ring.

Duality Pairs Induced by One-Sided Gorenstein Subcategories

“…Note that for a commutative noetherian ring R and a semidualizing bimodule R C R , it was proved in [19,Theorem 3.11] that R is artinian if and only if (GP C (R), GI C (R)) is a (coproduct-closed) duality pair.…”