First, second, and third change of rings theorems for the Gorenstein homological dimensions

TL;DR

This paper extends classical change of rings theorems to Gorenstein homological dimensions over arbitrary rings, broadening their applicability beyond Noetherian rings using strongly Gorenstein modules.

Contribution

It introduces generalized change of rings theorems for Gorenstein projective and injective dimensions over arbitrary rings, expanding prior results limited to Noetherian rings.

Findings

Extended first, second, third change of rings theorems to Gorenstein dimensions

Generalized results from Noetherian to arbitrary rings

Utilized strongly Gorenstein modules for proofs

Abstract

Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established, in this paper, for the Gorenstein projective dimension is a generalization of a result established over Noetherian rings and for finitely generated modules. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this paper for the Gorenstein projective dimension is a generalization of a $G$-dimension of a finitely generated module $M$ over a noetherian ring $R$.