Let R be an associative ring with identity. It is shown that every Σ-cotorsion left R-module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ-cotorsion. It follows that, for countable R, the class of Σ-cotorsion modules is closed under elementary equivalence and pure submodules. The modules M that satisfy this descending chain condition are the cotorsion analogues of totally transcendental modules; we characterize them as the (...) modules M for which , for every countably presented flat module F. (shrink)

We study Martsinkovsky–Russell torsion modules with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the Σ-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative Σ-pure-injective modules extends the classical characterization of Gruson and Jenson as well as Zimmermann. 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). (shrink)

Let R be an associative ring with identity. It is shown that every Σ-cotorsion left R-module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ-cotorsion. It follows that, for countable R, the class of Σ-cotorsion modules is closed under elementary equivalence and pure submodules. The modules M that satisfy this descending chain condition are the cotorsion analogues of totally transcendental modules; we characterize them as the (...) modules M for which, for every countably presented flat module F. (shrink)