Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, Formula: see text-submodules and Formula: see text-epimorphic images where Formula: see text is either the class of embeddings, RD-embeddings or pure embeddings. We show that the Formula: see text-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and Formula: see text-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.