Cofinitely Generalized Weak Supplemented Modules

Abstract

Let R be a ring and M be a left R-module. M is called cofinitely generalized weak supplemented (or briefly CGWS-module) if every cofinite submodule of M has a generalized weak supplement in M. In this paper, we give various properties of that kind of modules. It is shown that (1) A module M is CGWS-module if and only if every maximal submodule has a generalized weak supplement in M. (2) The class of cofinitely generalized weak supplemented modules are closed under taking homomorphic images, arbitrary sums, generalized covers and short exact sequences. (3) A ring R is semilocal if and only if every left R-module is a CGWS-module. (4) A commutative domain R is h-semilocal if and only if every torsion R module is cofinitely generalized weak supplemented.