J-semi regular modules

Abstract Let R be a ring with identity and let M be a left R-module. M is called J-semiregular module if every cyclic submodule of M is J-lying over a projective summand of M, The aim of this paper is to introduce properties of J-semiregular module Especially, we give characterizations of J-semiregular module. On the other hand, the notion of J-semi hollow modules is studied as a generalization of semi hollow modules, finally F-J-semiregular modules is studied as a generalization of F-semiregular modules.