Pure Baer injective modules

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