Gorenstein homological properties of tensor rings

TL;DR

This paper investigates how Gorenstein homological properties transfer between a noetherian ring and its associated tensor ring, providing characterizations of Gorenstein modules in this context.

Contribution

It establishes conditions under which Gorenstein properties are preserved between a ring and its tensor ring, and characterizes Gorenstein projective modules over the tensor ring.

Findings

$R$ is Gorenstein if and only if $T_R(M)$ is Gorenstein under certain conditions

Characterization of Gorenstein projective $T_R(M)$-modules in terms of $R$-modules

Conditions for Gorenstein property transfer between $R$ and $T_R(M)$

Abstract

Let $R$ be a two-sided noetherian ring and $M$ be a nilpotent $R$-bimodule, which is finitely generated on both sides. We study Gorenstein homological properties of the tensor ring $T_R(M)$. Under certain conditions, the ring $R$ is Gorenstein if and only if so is $T_R(M)$. We characterize Gorenstein projective $T_R(M)$-modules in terms of $R$-modules.