  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 ring is right semihereditary. 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. As a corollary, we show that the class of torsion abelian groups
with pure embeddings is not superstable.
