⊕-J-supplemented modules

Abstract

Abstract Let R be an associative ring with identity and let M be a unitary left R-module. In this paper, we introduce a ⊕-Jacobson-supplemented module (for short ⊕-J-supplemented), as a generalization of J-supplemented module. We state the main properties of a ⊕-J-supplemented modules and supplying by examples and remarks for these module and several properties of this modules are given. Also we introduce weakly ⊕-J-supplemented modules and give a characterization of weakly ⊕-J-supplemented modules a conditions under which the direct sum of weakly ⊕-J-supplemented modules is weakly ⊕-J-supplemented are given. We define a cofinitely ⊕-J-supplemented modules (for short, cof-⊕-J-supplemented) and some type of modules that are related to cof-⊕-J-supplemented module and prove properties of this type of modules. Also we discuss the relation between them with examples and remarks.