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.