Pure Injective Modules Relative to Torsion Theories

Abstract

Let be a hereditary torsion theory on the category Mod-R of right R-modules. A right R-module M is called pure -injective if it is injective with respect to every pure exact sequence having a -torsion cokernel. Every module has a pure -injective envelope. A module M is called purely quasi -injective if it is fully invariant in its pure -injective envelope. A module M is called quasi pure -injective if maps from dense pure submodules of M into M are extendable to endomorphisms of M. The class of pure -injective modules is properly contained in the class of purely quasi -injective modules which is in turn properly contained in the class of quasi pure -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.