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In this paper, three types of rings were reviewed: invo-clean, invo-t-clean and invo-k-clean, the ring invo-clean is invo-t-clean and invo-k-clean. Since invo-t-clean ring is invo-k-clean when k=3. Various examples representing elements of the three rings were presented, as were other rings in Zn that satisfy the three types, especially for the invo-k-clean ring, where different examples of rings were taken at different values of k. There are properties that all rings share and others that differ among them, the most important of which is characteristic, where the invo-clean ring is 24, and the invo-t-clean ring is 120 while invo-k-clean there is difficulty in proving it.

If for any maximal right ideal P of B and a ∈ N(B) ,aB/ aP is almost N-injective, then a ring B is said to be right generalized almost N-injective. In this article, we present some significant findings that are known for right almost N-injective rings and demonstrate that they hold for right generalized almost N-injective rings. At the same time, we study the case in which every S.S.Right B-module is generalized almost N-injective.

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.

An ideal K of a ring R is called right (left) generalized pure (GP -ideal) if for every a ∈ K, there exists m ∈ Z+, and b ∈ K such that am = am b ( am = b am) . A ring R is called MLGP-ring if every right maximal ideal is left GP-ideal. In this paper have been studied some new properties of MLGP-rings and the relation between this rings and strongly π-regular rings some of the main result of the present work are as follows: 1- Let R be a local ,MLGP and SXM ring. Then: (a) J (R) = 0. (b) If R is NJ-ring. Then r(am) is a direct sum and for all ∈ R, m ∈ Z+. 2- Let R be a local, SXM and NJ-ring . Then R is strongly π-regular if and only if R i LGP.

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.

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 called completely right YJ-injective (briefly, right CYJ injective ) if every homomorphic image of R is right YJ-injective. In this paper, we study completely right YJ-injective rings and their connection with Von Neumann regular rings. In addition, we also study regularity of rings whose ring homomorphic images are right YJ-injective as right R-modules

As a generalization of right weakly regular rings, we introduce the notion of right n-weakly regular rings, i.e. for all aN(R), aaRaR. In this paper, first give various properties of right n-weakly regular rings. Also, we study the relation between such rings and reduced rings by adding some types of rings, such as NCI, MC2 and SNF rings.

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 p-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 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) sp-weakly regular if for each a Î R and a positive integer n, aÎ aR aR (aÎ R aR a). In this paper, we continue to study sp-weakly regular rings due to R. D. Mahmood and A. M. Abdul-Jabbar [8]. We first consider properties and basic extensions of sp-weakly regular rings, and we give the connection of sp-weakly regular, semi p-regular and p-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 Î R, aÎ aR aR, for some positive integer n. Such rings are called sp-weakly regular rings and give some of their basic properties as well as the relation between sp-weakly regular rings, strongly p-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.

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.