Wa-Module

Abstract

M.A.Hassin and A.B.Hussien introduced thefollowing concept :(a,H) is called aringoverG if H is asubgroup of groupG and aϵG and a has finite order or infinite order ,  we called the ring  (w_a,+,.) ; (a,H) Ring over G where

w_a= {a^mHaⁿ,m,nϵZ ,aϵG\H }

with two binary operation + and. Such that

 a^m1 Haⁿ¹ ,a^m2Haⁿ²Ïµ wa , m₁,m₂,n_1,n₂ ϵZ

1-a^m1Haⁿ¹+a^m2Haⁿ² =a^m1+m2 Ha^{n1+n2   .                                     Â} 

2-a^m1Ha^n1  .a^m2Haⁿ²  =a^m1+m2 Ha^{n1+n2  .                                      Â} 

In [4] , (w_a,+,.) is commutative ring with unity  aHa these lead us to give the definition of  w_a-module define on the ring   which is for   any commutative ring with unity element.

         The main purpose of this work is to give definition of wa-module and some  properties of W_a module many new and useful results are

Obtain about this concept, and we illustrate that by examples.