On SNF-rings , I

A ring R is called right SNF-rings if every simple right R-module is N-flat . In this paper , we give some conditions which are sufficient or equivalent for a right SNF-ring to be n-regular (reduced) .It is shown that

, we give some conditions which are sufficient or equivalent for a right SNF-ring to be n-regular (reduced) .It is shown that 1-If ) (a r is a GW-ideal of R for every R a  . then , R is reduced if and only if R is right SNFring.

Introduction
Throughout this paper, R denotes an associative ring with identity and all modules are unitary .We write ) (R J J = for the Jacobson radical of R , and ) ) for the right (left) singular ideal of R . The right and left annihilators of a subset X of a ring R are written as ) ( X r and ) ( X l . A right R-module M is said to be flat if , given any monomorphism Q is also monomorphism [1] . Generalizations on right flat modules have been studied by many authors (see [9] and [3]) . In [5] SF rings are defined and studied . A ring R is called right (left) SF-ring if every simple right (left) R-module is flat . In [9] , Wei and Chen first introduced and characterized a right N-flat modules , and gave many properties . A right R-module is called N-flat , if for any is the inclusion mapping . Actually , many authors investigated some properties of rings whose every simple right R-module is N-flat [4] and [9] .
Recall that a ring R is called reduced ring if it has no non zero nilpotent elements , or equivalently , 0 2 = a , that implies 0 = a for all R a  . A ring R is called reversible  [6,7] . Clearly , a reduced ring is right nil-injective , right NPP and n-regular ring [6] .

SNF-ring
The following lemma , which is duo to [9] , plays a central role in several of our proofs .

Lemma 2.1 :
1-Let B be a right R-module and there exists R-short exact sequence 3-Let R be a ring then, R is n-regular ring if and only if every right R-module is N-flat . Following [5] , a ring R is called MERT ring if every maximal essential right ideal is a two-sided ideal of R.
Clearly , a right SF-ring is right SNF-ring , but the converse is not true. Because there exists a reduced MERT ring which is not regular, there exists a reduced MERT ring R which is not right SF by [12,Theorem 1] . On the other hand ,by [9,Theorem 4.7], reduced ring is right SNF,so there exists a right SNF-ring which is not right SF [9].

Examples (3) :
1-Let 2 Z be the ring of integer modulo 2 and let } 1 : is reduced nregular ring and SF-ring ,therefore it is SNF-ring .

2-Let 2
Z be the ring of integer modulo 2 ,then 3-The ring of integers Z is SNF-ring but not SF-ring .

Lemma 2.2: [2]
Let R be a reversible ring , then

Proposition 2.3 :
On SNF-rings , I 15 Let R be a right N duo, SNF-ring , then 0 Proof : contains a non-zero element a such that . This proves that ) (a l is a right ideal of R . Therefore , there exists a maximal right ideal of  , which implies 0 = a .Thus , R is a reduced ring . ■ Next, we recall the following result of Wei and Chen [6] which proved the link between nil-injective and n-regular rings .

Theorem 2.8 :
The following conditions are equivalent for a ring R 1-R is a n-regular ring .

2-Every left R-module is nil-injective . 3-Every cyclic left R-module is nil-injective . 4-
R is left nil-injective left NPP ring . From Theorems (2.4 and 2.8) and Lemma (2.1) , we get the following theorem .

Theorem 2.19 :
Let R be a reversible ring. Then , R is a right SNF-ring ,if and only if R is nilinjective .■

3-Rings whose simple singular right R-module are N-flat
In this section , we give an investigation of several properties for rings whose simple singular right R-modules are N-flat . Also , we study the relations between such rings and weakly regular ring .

Definition 3.1 :
A ring R is said to be right SSNF-ring , if every simple singular right R-module is N-flat .

Theorem 3.2 :
, which is a contradiction . Therefore , which is a contradiction . Therefore 0 . Following the proof of (1) we get , which is a required contradiction . Therefore , 0 [11] if , for any right ideal I isomorphic to a complement right ideal of R , every right R-homomorphism of I into R extends to an endomorphism of R . In [11] , shows that if R is right GQ-injective ring , then is regular . The next result is considered a necessary and sufficient condition for SSNF-rings to be regular ring .

Theorem 3.3 :
Let R be reversible ring .Then , the following statements are equivalent : 1) R is regular ring 2) R is a right GQ-injective ring and right SSNF-ring . Proof : 1 → 2 Observe that if R is regular then R is n-regular and so every right R-module is N-flat by [9,Theorem 4.2] .So we are done .
and R is regular ring . ■ Following [7], a ring R is called strongly min-able if every right minimal idempotent element is left semicentral .

Theorem 3.4 :
Let R be a strongly right min-able , MERT ring . If R is right SSNF-ring , then R is a right weakly regular ring .

Proof :
We Hence , R is a weakly regular ring . ■