Gorenstein objects in the n-Trivial extensions of abelian categories

TL;DR

This paper introduces a new categorical concept of Gorenstein objects in abelian categories, explores their transfer to n-trivial extensions, and applies findings to module categories and triangular matrix rings.

Contribution

It defines Gorenstein objects in abelian categories and characterizes their behavior in n-trivial extensions, providing new insights into their structure and applications.

Findings

Gorenstein objects are characterized in n-trivial extensions.

Transfer properties of Gorenstein objects are established.

Applications to modules over rings and triangular matrix rings are demonstrated.

Abstract

Given an abelian category, we introduce a categorical concept of (strongly) Gorenstein projective (resp., injective) objects, by defining a new special class of objects. Then we study the transfer of these properties when passing to an abelian category and its n-trivial extension category and also give a characterization of Gorenstein object over it. We give, at the end, applications of this study on the category of modules over an associative ring and triangular matrix rings.