Injective generation for graded rings

TL;DR

This paper explores the concept of injective generation in graded rings, establishing conditions and methods for various classes including tensor and trivial extension rings, with implications for module category theory.

Contribution

It introduces new conditions for injective generation in graded rings and applies these to tensor, trivial extension, and twisted tensor product rings, expanding theoretical understanding.

Findings

Injective generation relates closely to graded injective generation.

Necessary and sufficient conditions are established for tensor and trivial extension rings.

Injective generation is proven for twisted tensor products of finite-dimensional algebras.

Abstract

In this paper we investigate injective generation for graded rings. We first examine the relation between injective generation and graded injective generation for graded rings. We then reduce the study of injective generation for graded rings to the study of injective generation for certain Morita context rings and we provide sufficient conditions for injective generation of the latter. We then provide necessary and sufficient conditions so that injectives generate for tensor rings and for trivial extension rings. We provide two proofs for the class of tensor rings, the one uses covering theory and the other uses the framework of cleft extensions of module categories. We finally prove injective generation for twisted tensor products of finite dimensional algebras.