$\mathcal{X}$-Gorenstein projective and $\mathcal{Y}$-Gorenstein injective modules over tensor rings

TL;DR

This paper characterizes $ ext{Gorenstein}$ projective and injective modules over tensor rings, providing explicit conditions and applications to trivial ring extensions and Morita context rings.

Contribution

It offers new characterizations of Gorenstein modules over tensor rings, extending the understanding of module properties in this algebraic context.

Findings

Characterization of $Ind( ext{X})$-Gorenstein projective modules over tensor rings.

Explicit description of $ ext{Y}$-Gorenstein injective modules over tensor rings.

Applications to trivial ring extensions and Morita context rings.

Abstract

Let $T_R(M)$ be a tensor ring and $\mathcal{X}$, $\mathcal{Y}$ be two classes of $R$-modules. Under certain conditions, we prove that a $T_R(M)$-module $(A, u)$ is $Ind(\mathcal{X})$-Gorenstein projective if and only if $u$ is monomorphic and $coker(u)$ is an $\mathcal{X}$-Gorenstein projective $R$-module. $\mathcal{Y}$-Gorenstein injective $T_R(M)$-modules are also explicitly described. As a consequence, the characterizations of Ding projective and Ding injective modules over $T_R(M)$ are obtained. Some applications to trivial ring extensions and Morita context rings are given.