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