Generalization of T-small submodules

Abstract

Let R be associative ring with identity and let M be unitary left R-module.  A submodule N of M is called, T - small in M denoted by N ≪_T M, in case for any submodule X ⊆ M, T ⊆ N + X implies that T ⊆ X.  In this paper, we introduce the concept of GT - small submodule in M.  A submodule N of an R-module M is called GT-small submodule in M, denoted by N ≪_GT M, in case for every essential submodule X of M, T ⊆ N + X implies that T ⊆ X.  We introduce and study the concepts GT-hollow module, GT-lifiting modules and GT-supplement submodules as a generalization of T-hollow module, T-lifiting moules and T-supplement submodules respectively we supply some examples and properties of these modules.