Gorenstein dimensions in trivial ring extensions

TL;DR

This paper investigates the behavior of Gorenstein dimensions in trivial ring extensions, revealing that the Gorenstein global dimension can often be infinite and exploring the transfer of Gorenstein properties between rings and their extensions.

Contribution

It provides new insights into the Gorenstein dimensions in trivial ring extensions and highlights limitations in transferring Gorenstein properties in pullback constructions.

Findings

Gorenstein global dimension often infinite in trivial ring extensions

Transfer of Gorenstein properties between rings and extensions is limited

Counterexample shows Gorenstein projective transfer failure in pullbacks

Abstract

In this paper, we show that the Gorenstein global dimension of trivial ring extensions is often infinite. Also we study the transfer of Gorenstein properties between a ring and its trivial ring extensions. We conclude with an example showing that, in general, the transfer of the notion of Gorenstein projective module does not carry up to pullback constructions.