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A ring N is known as a right (left) SGmF – ring, if each simple right (left) N -modules are Gm – Flat. In this paper, we investigate characterizations of SGmF – ring. And from the results: 1. If N is a semi-prime, and right SGmF–ring, then the C(N) is regular ring. 2. If N is SGmF – ring, with every ideal of N is principal then: a. N is NJ iff N is feckly reduced, b. N is m – regular ring, if N is ZC ring.

A semi group K is defined to be right (left) weakly regular if for all a ∈ K, a ∈ aKaK (a ∈ KaKa). In this paper, we show the following results: Let K be a W- regular semi group. Then: Any AF inter...

An ideal I of a ring R is said to be right (left) Pure if for every , there is such that . A ring R is said to be right (left) MP-ring, if every maximal right (left) ideal of R is a left (right) pure. In this paper have been studied some new properties of MP-rings, there connections with strongly regular rings. Some of the main result of the present work are as follows: 1- Let R be aright MP-ring, r(a) is a W-ideal for all then a- Every essential ideal is a direct summand. b- R is strongly regular ring. 2- Let R be aright MP-ring. If R is right almost abelian left NBF ring, then R is strongly regular.

In this paper, we investigate some properties of rings whose simple singular right R- modules are A Gp-injective (or SSAGP- injective for short). It is proved that: Y(R)=0 where R is a right SSAGP- injective rings. It is also proved that Let R be a complement right bounded, SSAGP – injective rings and every maximal essential right ideal is Gw-ideal. Then R is strongly regular ring. Let R be SSAGP – injective and r(e) is Gw-ideal for every idempotent element . Then Z(R)=0. Let R be SSAGP – injective, MERT and right CM. Then R is either strongly regular or semi simple Artinian.

The ring R is called right almost weakly np –injective, if for any , there exists a positive integer n and a left ideal of R such that . In this paper, we give some characterization and properties of almost weakly np – injective rings. And we study the regularity of right almost weakly np – injective ring and in the same time, when every simple (simple singular) right R – module is almost weakly np – injective, we also give some properties of an R.

In this paper, we continue studies of several other authors, on QⱤ (quasi-regular) and JCPI (JCP-injective) rings. In particular, we investigate some characterizations of several basic properties of QⱤ and JCPI rings. Also, we prove that, if Ɽ is a semisimple and quasi duo, then Ɽ is strongly regular if and only if every simple singular right Ɽ-modules are JCPI.

As a popularization of weakly π-regular rings, we tender the connotation of W P Wπ-regular rings, that is if for each 𝔞 ∈ Ɉ(𝔑), there exist a natural number 𝔫 such that 𝔞𝔫 ∈ 𝔞𝔫 𝔑 𝔞𝔫 𝔑 (𝔞𝔫 ∈ 𝔑 𝔞𝔫 𝔑 𝔞𝔫). In this treatise, numerous properties of this sort of rings are discussed, some important results are secured. Using the connotation of P Wπ-regular rings. It is show that : 1- Let 𝔑 be a right P Wπ-regular ring and ℵɈ-rings with 𝔞𝔫𝔑 = 𝔑𝔞𝔫 for every 𝔞 ∈ Ɉ(𝔑) and for at least one of a natural number. Then Ɉ(𝔑) = ℵ𝔑. 2- Let 𝔑 a right P Wπ-regular ring and 𝔞𝔑 = 𝔑𝔞 for each ∈Ɉ(𝔑). Then 𝔑 is right P .𝔗-ring. 3 Let 𝔑 be a ring with ɍ(𝔞) ⊆ ɭ(𝔞), for each ∈ Ɉ (𝔑). If any of the next conditions are hold, then 𝔑 is P Wπ-regular rings : i – Every maximal right ideal of 𝔑 is a right annihilator and right Ɉ PP -ring. ii – any simple singular right 𝔑-module is Ɉ-injective and 𝔑 is semi prime.

A ring 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 1- If is a GW-ideal of R for every . then ,is reduced if and only if is right SNF-ring. 2- If is an reversible, then is regular if and only if is right GQ-injective and SSNF-ring .

As a generalization of right injective rings, we introduce the nation of right injective rings, that is for any right nonsingular element of R, there exists a positive integer and and any right - homomorphism , there exists such that for all . In this paper, we first introduce and characterize a right injective rings . Next , connection between such ring and quasi regular rings and weakly regular rings.

Let I be a right ideal of a ring R , then R/I is right N-flat module if and only if for each , there exists and a positive integer n such that and .In this paper, we first introduce and characterize rings whose every simple singular right R-module is N - flat. Next, we investigate the strong regularity of rings whose every simple singular right R - module is N-flat. It is proved that : R is strongly regular ring if and only if R is a wjc , MERT and 2 - primal ring whose simple singular right R- module is N - flat. Let R be a wjc ring satisfying condition (*). If every simple singular right R-module is N-flat .Then, the Center of R is a regular ring.

A ring R is said to be generalized right simple singular AP-injective, if for any maximal essential right ideal M of R and for any bÎM, bR/bM is AP-injective. We shall study the characterization and properties of this class of rings. Some interesting results on these rings are obtained. In particular, conditions under which generalized simple singular AP-injective rings are weakly regular rings, and Von Neumann regular rings.

As a generalization of regular rings, that is a ring is called n-regular if for all . In this paper, we first give various properties of n-regular rings. Also, we study the relation between such rings and reduced rings by adding some types of rings, such as NCI rings, and other types of rings.

Let I be a right ideal of R, then R / I is a right N–flat if and only if for each a Î I, there exists b Î I and a positive integer n such that an ≠ 0 and an = ban. In this paper, we first give and develop various properties of right N-flat rings, by which, many of the known results are extended. Also, we study the relations between such rings and regular, p-biregular ring.

The purpose of this paper is to study the rings in which every maximal (essential) right ideal is a left GP- ideal. Such rings will be called right MGP- rings (EGP- rings). We give the basic properties of such rings and their connection with strongly - regular rings, fully left idempotent rings, and - weakly regular rings.

A right module M is called Wnil-injective if for any 0 ¹ a Î N( R ), there exists a positive integer n such that an ¹ 0 and any right R-homomorphism f : anR ® M can be extended to R ® M. In this paper , we first give and develope various properties of right Wnil-injective rings, by which, many of the known results are extended. Also, we study the relations between such rings and reduced rings by adding some types of rings, such as SRB-rings, and other types of rings.

In this paper we investigate von Neumann regularity of rings whose simple singular right R-modules are flat. It is proved that a ring R is strongly regular if and only if R is a semiprime right quasi-duo ring whose simple singular right R-modules are flat. Moreover, it is shown that if R is a P. I.- ring whose simple singular right R-modules are flat, then R is a strongly π- regular ring. Finally, it is shown that if R is a right continuous ring whose simple singular right R-modules are flat, then R is a von Neumann regular ring.

In this work we give some new properties of GP- ideals as well as the relation between GP- ideals, π - regular and simple ring. Also we consider rings with every principal ideal are GP- ideals and establish relation between such rings with strongly - π regular and local rings.

In this paper, we study rings whose principal right ideals are left pure. Also we shall introduce the concept of a fuzzy bi-ideal in a ring, and give some properties of such fuzzy ideals. We also give a characterization of whose principal right ideal are left pure, fuzzy duo ring in terms of fuzzy deals.

A ring R is said to be right(left) sπ-weakly regular if for each a ∈ R and a positive integer n, a n ∈ a n R a n 2 R (a n ∈ R a n 2 R a n ). In this paper, we continue to study sπ-weakly regular rings due to R. D. Mahmood and A. M. Abdul-Jabbar (8). We first consider properties and basic extensions of sπ-weakly regular rings, and we give the connection of s π-weakly regular, semi π-regular and π-biregular rings.

The purpose of this paper is to study the class of the rings for which every maximal right ideal is left GP-ideal. Such rings are called MGP-rings and give some of their basic properties as well as the relation between MGP-rings, strongly regular ring, weakly regular ring and kasch ring.

The purpose of this paper is to study a new class of rings R in which, for each a ∈ R, a n ∈ a n R a n 2 R, for some positive integer n. Such rings are called s π-weakly regular rings and give some of their basic properties as well as the relation between s π-weakly regular rings, strongly π-regular rings and division rings.

This paper introduces the notion of a right MEP-ring (a ring in which every maximal essential right ideal is left pure) with some of their basic properties; we also give necessary and sufficient conditions for MEP – rings to be strongly regular rings and weakly regular rings.

The purpose of this paper is to study GP-injective modules and give some of it is properties. Also, we proved: (1) If every simple right R-module is GP-injective, and R is reversible ring, then R is a right weakly -regular. (2) Let R be a right weakly regular, right quasi-duo ring, if every simple singular right R-module are GP-injective, then R is -regular.

In this work we consider weakly regular rings whose simple singular right R-Modules are flat. We also consider the condition (*): R satisfies L(a)Ír(a) for any aÎR. We prove that if R satisfies(*) and whose simple singular right R-module are flat, then Z (R) the center of R is a von Neunann regular ring. We also show that a ring R either satisfies (*) or a strongly right bounded ring in which every simple singular right R-module is flat, then R is reduced weakly regular rings.

This paper introduces the notion of maximal GP-ideal .We studied the class of rings whose maximal left ideal are right GP-ideal. We call such ring MRGP-rings. We consider a necessary and sufficient condition for MRGP-rings to be MRCP-rings. We also study the connection between MRGP-ring, kasch ring, division ring and the strongly regular ring.

In this work, we study generalized simple P-injective rings. New properties of such rings are given, and a characterization of division rings and strongly regular rings in terms of GSP rings is obtained.

The objective of this paper is to initiate the study of rings in which each simple right R-module is flat, such rings will be called right SF-rings. Some important properties of right SF-rings are obtained. Among other results we prove that: If R is a semi prime ERT right SF-ring with zero socle, then R is a strongly regular ring.