Relative injective modules, superstability and noetherian categories

TL;DR

This paper explores the properties of various classes of modules, establishing a connection between model-theoretic superstability and algebraic noetherian categories, and characterizing key ring classes through this relationship.

Contribution

It introduces a model-theoretic approach to characterize noetherian categories and rings using superstability, extending classical module theory results.

Findings

$ ext{M}$-injective modules satisfy a Baer-like criterion

Superstability is equivalent to a noetherian category for these classes

Characterizations of rings like noetherian, pure semisimple, and artinian rings

Abstract

We study classes of modules closed under direct sums, $\mathcal{M}$-submodules and $\mathcal{M}$-epimorphic images where $\mathcal{M}$ is either the class of embeddings, $RD$-embeddings or pure embeddings. We show that the $\mathcal{M}$-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, $RD$-injective modules, pure injective modules, flat cotorsion modules and $\mathfrak{s}$-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah's non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.