A note on torsion modules with pure embeddings

Abstract

We study Martsinkovsky-Russell torsion modules [MaRu20] with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pureinjective and the Σ-pure-injective modules relative to the class of torsion modules assuming that the ring is right semihereditary. Our characterization of relative Σ-pure-injective modules extends the classical charactetization of [GrJe76] and [Zim77, 3.6].We study the limit models of the class and determine when the class is superstable assuming that the ring is right semihereditary. As a corollary, we show that the class of torsion abelian groups with pure embeddings is strictly stable, i.e., stable not superstable.