The category of modules on an n-trivial extension: the basic properties

TL;DR

This paper explores the categorical properties of n-trivial extensions of rings by modules, characterizing key objects and applying results to ring perfection and self-injective dimensions.

Contribution

It introduces the concept of n-trivial extensions of categories by endofunctors and characterizes projective, injective, and flat objects within this framework.

Findings

Characterization of projective, injective, and flat objects in n-trivial extension categories.

Conditions under which n-trivial extension rings are k-perfect.

Results on the self-injective dimension of n-trivial extension rings.

Abstract

In this paper we investigate a categorical aspect of $n$-trivial extension of a ring by a family of modules. Namely, we introduce the right (resp., left) $n$-trivial extension of a category by a family of endofunctors. Among other results, projective, injective and flat objects of this category are characterized. We end the paper with two applications. We characterize when an $n$-trivial extension ring is $k$-perfect and we establish a result on the selfinjective dimension of an $n$-trivial extension ring.