(projectively coresolved) Gorenstein flat modules over tensor rings

TL;DR

This paper characterizes projectively coresolved Gorenstein flat modules over tensor rings, linking their properties to modules over the base ring and exploring applications to ring extensions and Morita contexts.

Contribution

It provides a new characterization of Gorenstein flat modules over tensor rings and describes a class of Gorenstein projective modules explicitly.

Findings

Characterization of Gorenstein flat modules over tensor rings.

Explicit description of Gorenstein projective modules.

Applications to trivial ring extensions and Morita rings.

Abstract

Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize projectively coresolved Gorenstein flat modules over $T_R(M)$, showing that a $T_R(M)$ module $(X,u)$ is projectively coresolved Gorenstein flat if and only if $u$ is monomorphic and $coker(u)$ is a projectively coresolved Gorenstein flat $R$-module. A class of Gorenstein at modules over $T_R(M)$ are also explicitly described. We discuss applications to trivial ring extensions and Morita context rings.