Gorenstein homological dimensions of modules over triangular matrix rings

TL;DR

This paper investigates the Gorenstein homological dimensions of modules over triangular matrix rings, providing characterizations and conditions for strongly Gorenstein projective and injective modules.

Contribution

It offers new characterizations of Gorenstein homological dimensions and conditions for strong Gorenstein properties over triangular matrix rings.

Findings

Characterization of Gorenstein homological dimensions over T

Conditions for strongly Gorenstein projective modules

Conditions for strongly Gorenstein injective modules

Abstract

Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.