A note on torsion modules with pure embeddings

TL;DR

This paper investigates torsion modules with pure embeddings using model theory, characterizing pure-injective modules, analyzing limit models, and establishing the stability properties of the class of torsion abelian groups.

Contribution

It provides a new model-theoretic characterization of pure-injective and $\Sigma$-pure-injective torsion modules, extending classical results and analyzing stability conditions.

Findings

Characterization of pure-injective and $\Sigma$-pure-injective modules relative to torsion modules

Determination of conditions for the class to be superstable

The class of torsion abelian groups with pure embeddings is strictly stable, not superstable

Abstract

We study Martsinkovsky-Russell torsion modules [MaRu20] with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the $\Sigma$-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative $\Sigma$-pure-injective modules strictly extends the classical charactetization of [GrJe76] and [Zim, 3.6]. We study the limit models of the class and determine when the class is superstable assuming that the torsion submodule is a pure submodule. As a corollary, we show that the class of torsion abelian groups with pure embeddings is strictly stable, i.e., stable not superstable.