Pure injective modules relative to torsion theories

Let τ be a hereditary torsion theory on the category Mod-R of right R-modules.A right R-module M is called pure τ -injective if it is injective with respect to every pure exact sequence having a τ -torsion cokernel.Every module has a pure τ -injective envelope.A module M is called purely quasi τ -injective if it is fully invariant in its pure τ -injective envelope.A module M is called quasi pure τ -injective if maps from τdense pure submodules of M into M are extendable to endomorphisms of M .The class of pure τ -injective modules is properly contained in the class of purely quasi τ -injective modules which is in turn properly contained in the class of quasi pure τ -injectives.A torsion theoretic version of each of the concepts of regular and pure semisimple rings is characterized using the above generalizations of pure injectivity.