• σ- PURITY AND σ- REGULAR RINGS AND MODULES

Abstract

The aim of this paper is to relativize the concept of  purity and σ - purity defined and studied by Azumaya [6] with respect to an arbitrary hereditary torsion theory given by a left exact torsion radical  and also relates these concepts with the notions of  purity as given by B. B. Bhattacharya and D. P. Choudhury [7]. We also develope the theory of  purity and  purity relative to a torsion theory with radical  where  is a finitely generated or cyclic  module and  is an  matrix determined by a system of linear equations  where  (a left  module) for each  and  are unknowns, which is weaker than the usual purity and given a sufficient condition for these two coincide. In this present paper we relativize the concept of the   pure and  blatness of a module. We also discuss about  regular modules and weakly  regular modules and its inter relationship. We also discuss about finitely generated  blat modules and its condition for  projectivity in Noetherian ring.