Generalization of Local Modules

Abstract

     In this paper we introduce a generalization of local module. The notion of 2-Local module can be considered a new concept (generalization of local modules). Let T be a ring with identity.  2-maximal submodule can define as, any submodule B₁ of B is called 2-maximal if (B/B₁) is a 2-regular module.  2-Local module means every submodule A of a T-module B where    is a unique 2-maximal submodule. Almost any unique maximal submodule of a T-module  gives 2-local module.  Also, we can present a generalization of local module by, if  is a fully prime module over the ring  and every submodule B₁ of a T-module B is a unique almost max-submodule, this means  is a 2-local. Every semi-maximal submodule   over chained module  is maximal (2-maximal submodule) and hence  is 2-local module.  Some properties and examples of 2-Local module are given. Finally some implications of 3-local module have been presented.