2-Maximal Submodules and Related Concepts

Throughout this paper r represents commutative ring with identity and m is unitary left r-module, the purpose of this paper is to study new concept (up to our knowledge) , named 2-maximal submodule which is a generalization of maximal submodule , where a submodule n of an r-module m is called 2-maximal submodul of m if and only if m/n is 2-regular r-module . many characterizations and properties of 2-maximal submodules are given . moreover we studied the behavior of 2-maximal submodule in some classes of module . finally we give the sufficient condition 2-maximal submodules to be semi-maximal weak-maximal submodules are introduced