Gorenstein projective, injective and flat modules over trivial ring extensions

TL;DR

This paper characterizes Gorenstein projective, injective, and flat modules over trivial ring extensions using generalized compatible bimodules, providing explicit criteria and applications to Morita context rings.

Contribution

It introduces generalized compatible bimodules to characterize Gorenstein modules over trivial ring extensions, extending existing theory.

Findings

Characterization of Gorenstein projective modules via exact sequences.

Explicit criteria for Gorenstein injective and flat modules.

Application to Morita context rings with zero bimodule homomorphisms.

Abstract

We introduce the concepts of generalized compatible and cocompatible bimodules in order to characterize Gorenstein projective, injective and flat modules over trivial ring extensions. Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$ such that $M$ is a generalized compatible $R$-$R$-bimodule and $\textbf{Z}(R)$ is a generalized compatible $R\ltimes M$-$R\ltimes M$-bimodule. We prove that $(X,\alpha)$ is a Gorenstein projective left $R\ltimes M$-module if and only if the sequence $M\otimes_R M\otimes_R X\stackrel{M\otimes\alpha}\rightarrow M\otimes_R X\stackrel{\alpha}\rightarrow X$ is exact and coker$(\alpha)$ is a Gorenstein projective left $R$-module. Analogously, we explicitly characterize Gorenstein injective and flat modules over trivial ring extensions. As an application, we describe Gorenstein projective, injective and flat modules over Morita context rings with zero bimodule homomorphisms.