• 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.
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