Idempotent Reflexive rings whose every simple singular right module are YJ-injective

1Introduction Throughout in this paper is associative ring with identity and all modules are unitary. For a subset of , the left(right) annihilator of X in R is denoted by { } a X If X r X l = )). ( )( ( , we usually abbreviate it to We write for the Jacobson radical, the set of nilpotent elements,the nil radical (that means the sum of all nil ideals), prime radical (that means the intersection of all prime ideals) and left singular ideal respectively.A ring R is called if A ring R is called 2-primal if Let I be a right (left) ideal of R,then R/I is a right (left) N-flat if and only if for each there exists and positive integer n such that and A ring R is said to be right


1-Introduction
Throughout in this paper is associative ring with identity and all modules are unitary.For a subset of , the left(right) annihilator of X in R is denoted by , we usually abbreviate it to W e w r i t e for the Jacobson radical, the set of nilpotent elements,the nil radical (that means the sum of all nil ideals), prime radical (that means the intersection of all prime ideals) and left singular ideal respectively.A ring R is called if A ring R is called 2-primal if Let I be a right (left) ideal of R,then R/I is a right (left) N-flat if and only if for each t h e r e e x i s t s and positive integer n such that and A ring R is said to be right Clearly, NI ring is NCI [3].A ring R is said to be N duo if aR=Ra, for all [12].
A ring R is said to be reflexive if implies for A ring R is said to be right idempotent reflexive if implies for A right R-module M is called YJ-injective if for any there exists a positive integer n such that and every right R-homomorphism of into M extends to one of R into M [11].YJ-injectivity is also called GPinjectivity, by several authors [5].

Some properties of idempotent reflexive ring whose simple singular module is YJ-injective.
In this section we give some properties of right idempotent reflexive ring whose every simple singular right R-module is YJ-injective then R is semiprime, J(R)=0, and right weakly regular ring.Every semiprime ring is reflexive, and every reflexive ring is right idempotent reflexive, but Kim in [4], gives example of right and left idempotent reflexive ring but notsemiprimenor reflexive ring.The next theorem gives condition makes the idempotent reflexive ring implies to reflexive and semiprime ring.

Theorem 2.1
Let R be a ring whose every simple singular rightR-module is YJinjective.Then the following conditions are equivalent: 1-R is semiprime.2-R is reflexive ring.

The Fourth Scientific Conference of the College of Computer Science & Mathematics
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3-R is right idempotent reflexive. Proof:
it is clear.
We shall show there is no nilpotent ideal in R, if not, suppose there exists with , , that is mean , there exists a maximal right ideal M of R containing If is not essential, then where ,since , then , so since R is right idempotent reflexive ring, which is a contradiction.Therefore M is an essential right ideal of R,we get that .Hence, there exists appositive integer n=1 such that and any R-homomorphism of R into R/M extends to one of R into R/M, we define such that where .It is clear that f is well define right R-homomorphism, since R/M is YJ-injective, there exists s u c h t h a t since , we get that which is also contradiction.Therefore .This is shows that R is

Theorem 2.2
Let R be a right idempotent reflexive ring whose every simple singular rightRmodule is YJ-injective.Then R is left nonsingular.Proof:

Let
such that then either or not, if not, there exists a maximal right ideal M of R containing If is not essential, then where , since , then , so By the same method as in the proof of Theorem 2.1, we get that in particular there exists and such that since then that is mean is essential left ideal of R, so for all left ideal I of R, in special case we take then so there exists for some since , we get that therefore but is essential left ideal of R, it follows that that is mean Therefore R is left non singular.

Theorem 2.3
LetR be a right idempotent reflexive ring whose every simple singular rightRmodule is YJ-injective.Then R/ J(R) is N-flat left R-module.

Proof:
We shall to show that R/ J(R) is N-flat left R-module, if not, suppose there exists either or not, if not, there exists a maximal right ideal M of R containing If is not essential, then where , since ,then , so since R is right idempotent reflexive ring, which is a contradiction.Therefore M is an essential right ideal of R, we get that R/M is YJ-injective, there exists a positive integer n and such that any R-homomorphism of R into R/M extends to one of R into R/M,let such that where , f is well defineright R-homomorphism, since R/M is YJ-injective, there exists s u c h t h a t 1 + M = f ( )=(b+M)( +M)=b +M, 1+M= b+M, since we get that which is a contradiction.That is mean in particular there exists and such that set , Assist.Lecturer\ College of Engineering\ University of Mosul ] 147 [ regular ring if for each , there exists and positive integer n such that A ring R is said to be right weakly regular ring for each , there exists such that Call a ring R, S-weakly regular ring if for all Call a ring R NCI if N(R) is contain a non-zero ideal of R whenever