On W -regular rings

. As a popularization of weakly -regular rings , we tender the connotation of W -regular rings , that is if for each , there exist a natural number such that . In this treatise , numerous properties of this sort of rings are discussed , some important results are secured . Using the connotation of W -regular rings . It is show that : 1- Let be a right W -regular ring and -rings with and for at least one of a natural number . Then . 2- Let a right W -regular ring and for each . Then is right -ring . 3- Let be a ring with , for each . If any of the next conditions are hold , then is W -regular rings : i m aximal right right -module -injective and is .


Introduction .
Over this treatise, refers to an associative ring with identity and each module is unitary -module . We write ( ) , ( ) , and ( ) for the Jacobson radical , the right singular ideal and the set of nilpotent elements of , respectively . We use the contraction , for the left , right annihilator of in .
-injective rings were defined and discussed [5] , [10]. A ring is define as a rightinjective [10], whether each , . Recall that is known as a right (left ) weakly -regular (W -regular ) [7] , if every , there is a natural number such that . According to [4] is said to be -weakly regular ring if for any , . A ring is said to be reduced if ( )=0 [3]. is said to be if for each 0 there is a natural number such that , every reduced is but convers is not true [8] . A ring is define is semiprime ring if and only if it contains no non-zero nilpotent ideal [2] .
An element in the ring is said to be right (left) -element , if there is an idempotent element ȩ in such that and .
is known as a right ( left ) -ring . whether each element in is right ( left ) -element [1]. For example is -ring [1] .
In this treatise, we shall popularize the connotation of weakly -regular rings to W -regular , numerous properties of this sort of rings are discussed , little conditions under which W -regular are -ring, -regular , strongly regular rings will be given . Remark : Every weakly -regular ring is W -regular ring but the converse is not always true : Let be the ring of integer . Then ( ) = 0. Then is W -regular ring which is not W -regular ring. ) , implies that . Therefore and so ,implies that ( is reduced ) . Therefore , which yields . Hence is right W -regular, The conversely is clear .

The relevance among right W -regular and other rings
Following [10] , is called right -regular ring ( -regular ) . whether for each . Following [10] , is said to be right -ring . If is projective for each . In [10] we give the following lemma:

Lemma 3.4 :
Let be a ring .Then it is right -ring iff , ȩ is some idempotent element in , .

Proposition 3.5 :
If is right -ring , then . Proof : Assume that , . it is clear that is projective , then must be direct summand of . But , it then essential in , but this is contradiction. Therefore .

Lemma 3.6 :
Assume that is right -ring , , for each . Then is reduced .

Theorem 3.7 :
Let is right -ring , , for each , and any right maximal ideal of is a right annihilator .Then is W -regular .
Proof : Suppose that , we must show that . If it is not hold, then there is a right maximal ideal containing . If , for some , we have ( ) , which implies . Then , a contradiction . Therefore . In particular , with , and . Hence which proves is right Wregular .
Following [3], is called strongly regular ring , if for each , there is , .