Approximaitly Primary Submodules

Abstract

The study deals with the notion of an approximaitly primary submodules of unitary left R-module M over a commutative ring R with identity as a generalization of a primary submodules and approximaitly prime submodules, where a proper submodule N of an R-module M is called an approximaitly primary submodule of M, if whenever ay∈N, for a∈R, y∈M, implies that either y∈N+soc(M) or a^k M⊆N+soc(M) for some positive integer k of Z. Several characterizations, basic properties of this concept are given. On the other hand the relationships of this concept with some classes of modules are studied. Furthermore, the behavior of approximaitly primary submodule under R-homomorphism are discussed.