For Some Results of Semisecond Submodules

Let R be a commutative ring with unity and let B be a unitary R-module. Let א be a proper submodule of B, א is called semisecond submodule if for any r∈R, r≠0, n∈Z+, either rnא=0 or rnא=rא. In this work, we introduce the concept of semisecond submodule and confer numerous properties concerning with this notion. Also we study semisecond modules as a popularization of second modules, where an R-module B is called semisecond, if B is semisecond submodul of B.


Introduction
Let ℛ be a commutative ring with unity and let ℬ be a unitary ℛ -module.S.Yass in [1] introduced the notation of second submodule and second module where a submodule ℵ of an ℛ -module ℬ is called second submodule if for every r∈ℛ, r≠0, either rℵ =ℵ or rℵ =0 and a module ℬ is called semisecond if ℬ is semisecond submodule of ℬ.This definition leads us to introduce the notion of semisecond submodule and semisecond module as a generalization of second submodule and second module, where a submodule ℵ of an ℛ-module ℬ is called Semisecond if for every r∈ ℛ, r≠0, n∈Z+, either r n ℵ =0 or r n ℵ =rℵ and a module ℬ is Semisecond if ℬ is semisecond submodule of ℬ.The main aim of this work is to give basic properties of Semisecond submodules.Moreover, we survey the relationships between semisecond submodules and other submodules.Over this work we designate S.R.M. for submodule of an ℛ-module, for integral domain, for finitely generated, s.t. for such that and N.Z.for non-zero.
We notice that the provision M is faithful cannot be dropped from proposition (2.6) for instance: Consider the Z-module Z6, Z6 is F.G. multiplication Z-module but not faithful.
Hence, we have the following result.
Proof:-since ℛ ℵ is a maximal ideal, then ℛ/ ℛ ℵ is a field and by remark and example The following result shows the direct sum of two semisecond submodules under certain condition.
An R-module ℬ is rendering semiprime if (0) is a semiprime submodule of ℬ.
Note that the opposite of previous proposition is not hold in public for instance: -Take ℬ=Z as Z-module.ℬ is prime so it is semiprime.Let ℵ=<6> is semiprime, but N is not semisecond since for every r∈Z, r≠0, r 2 ℵ≠(0) and r 2 ℵ≠rℵ.

Reminiscence that a module ℬ is rendering Coprime if
for every proper submodule ℵ of ℬ, see [5].Equivalently ℬ is coprime module if and only if ℬ is second module, see [6, th.(2.1.6)].