On Jacobson â€“ Small Submodules
Let R be an associative ring with identity and let M be a unitary left Râ€“module. As a generalization of small submodule , we introduce Jacobsonâ€“small submodule (briefly Jâ€“small submodule ) . We state the main properties of Jâ€“small submodules and supplying examples and remarks for this concept . Several properties of these submodules are given . Also we introduce Jacobsonâ€“hollow modules ( briefly Jâ€“hollow ) . We give a characterization of Jâ€“hollow modules and gives conditions under which the direct sum of Jâ€“hollow modules is Jâ€“hollow . We define Jâ€“supplemented modules and some types of modules that are related to Jâ€“supplemented modules and introduce properties of this types of modules . Also we discuss the relation between them with examples and remarks are needed in our work.