• σ- PROJECTIVITY AND σ- SEMI-SIMPLICITY IN MODULES

Abstract

An exact sequence   is called pure (- copure) if any torsion (torsion free)          module is projective (injective) relative to it. Since  is closed under factors (sub-modules). In this situation Walker’s [23] criterion of Co-purity is also applicable. The notation of an  module  is pure projective (- copure injective) if and only if  ( for all . In particular  for all .We denote the torsion sub-module of  by  . Walker proved that the class of pure copure sequences form a proper class whenever  is closed under homomorphic images (sub-modules) of an  module  and if  is closed under factors (sub-modules) then for any purecopure sequence  if  () and hence in this case Walker’s purity copurity  coincides with the earlier notion of purity. We try to define a class of modules projective with respect to a torsion theory and to show that they are none other thanpure flat modules. Here we define two torsion theoretic generalizations of projective modules and one of them will be characterized as pure flat modules. Also the semisimple ring of Rubin [21] will be characterized in terms of divisibility and purity. We also study about divisible modules and co-divisible modules. we try to specify pure injective and pure projective modules and also we enumerate some properties of divisibility and co-divisibility as such to giving of characterization for exactness of a torsion theory in terms of it divisible and co-divisible  modules. Most of these results of the theorem are proved by Lambek [17] for  purity. In this present paper we try to relate the strongly  projectivity,   projective modules, torsion projective modules and also, pure flat module.  we try to give the inter relationship between torsion modules, divisible modules, co-divisible modules and semi-simplicity of the modules  for a hereditary torsion theory with radical