Pure Baer injective modules

In this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure Baer injective module. Some properties and some characterization of such modules are established. We also introduce two notions closely related to pure Baer injectivity; namely, the notions of a ∑ -pure Baer injective module and that of SSBI-ring. A ring R is an SSBI-ring if and only if every smisimple R -module is pure Baer injective. To investigate such algebraic structures we had to define what we call p -essential extension modules, pure relative complement submodules, left pure hereditary rings and some other related notions. The basic properties of these concepts and their interrelationships are explored, and are further related to the notions of pure split modules.