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.
