  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.
