S-pure Submodules

Purity in module categories and its different generalizations have been present in the literature since the 1960s. Apart from its special role in module theory, the importance of purity increased during the last two decades through the developments of model theory and the study of locally finitely presented categories (e.g., [2, 6]). In the module theory context, that will be our framework, a purity of a particular interest is purity with respect to a class of cyclic modules, that was introduced by Stenström [7]. Connected to that, the present paper deals with purity with respect to the class of simple modules, previously considered by the author in [3, 4]. We set some notation and terminology. Throughout the paper R denotes an associative ring with nonzero identity and Mod-R denotes the category of right R-modules. The injective hull of a right R-module A will be denoted by E(A). Consider a short exact sequence in Mod-R