Raida D. Mahmood

University of Mosul · College of Computer Science and Mathematics

A new class of rings are introduced which every element in the ring are sum of involution and tripotent elements. This class called involution t-clean ring which is a generalization of invo-clean rings and subclass of clean rings. Some properties of this class are investigate. For an application in graph theory, new graph is defined called t-clean...

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 interior ideal λ of K is, a) idempotent. b) λ* M ≤ λ ∪ M for any AF right ideal M of K. c) (λ* M) ∪ (M*λ) ≤ λ ∪ M for any AF interior ideal M of K. • λ*K*...

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-mod...

As a popularization of weakly π -regular rings, we tender the connotation of W P W π -regular rings, that is if for each 𝔞 ∈ Ɉ(𝔑), there exist a natural number 𝔫 such that 𝔞 𝔫 ∈ 𝔞 𝔫 𝔑 𝔞 𝔫 𝔑 (𝔞 𝔫 ∈ 𝔑 𝔞 𝔫 𝔑 𝔞 𝔫 ). In this treatise, numerous properties of this sort of rings are discussed, some important results are secured. Using the connotation of P...

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 presen...

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 𝑎 𝑚 = 𝑎 𝑚 𝑏 ( 𝑎 𝑚 = 𝑏 𝑎 𝑚 ) . 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...

A ring ℜ is called right simple ℑ-flat (or right SℑF-ring for short) if, for every simple right ℜ -module is ℑ-flat. In this paper, several properties of this class of rings are given, some interesting results are obtained. Right SℑF-rings, ℑא rings are studied, several conditions under which right SℑF-rings are reduced rings, strongly Ɣ-regular ri...

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 strongl...

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 1. Let R be a complement right bounded, SSAGP – injective rings and every maximal essential right ideal is Gw-id...

Let R be a ring. The ring is called right almost WJCP-injective. If for any , (right non singular) there exists a left ideal of such that. In this paper, we give some characterization and properties of almost WJCP-injective rings, which are proper generalization of JCP-injective ring and almost AP-injective ring. Then, we study the regularity of th...

In (Stanley and Zhou,1998), introduced a class of AGP-injective rings, following this a ring R is called right AGP-injective if, for any a∈R, there exists n>1 such that an ≠0 and Ran is a direct summand of ℓr(an). In this paper we give a generalization of right AGP-injective rings, we introduced the notion of right weakly AGP-injective rings, that...

Let R be a ring. The ring is called right almost WJCP-injective. If for any , (right non singular) there exists a left ideal of such that. In this paper, we give some characterization and properties of almost WJCP-injective rings, which are proper generalization of JCP-injective ring and almost AP-injective ring. Then, we study the regularity of th...

Mathematics

In this paper, we continue the studies of several other authors, on nil-injective rings. In particular, we investigate some characterizations and several basic properties of these rings and the relationship between them and n-egular rings, SF-rings, the IN-rings and Kasch rings, respectively. Throughout this paper R denoted an associative ring with...

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 (si...

A right module M is called Wnil-injective if for any 0 ≠ a ∈ N(R), there exists a positive integer n such that a n ≠ 0 and any right R-homomorphism f : a n R → 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...

Let R be a ring. The ring R is called right Nil-pure, if for any í µí²‚ ∈ í µí±¹ , r(a) is an aleft pure ideal of R. In this paper, we give some characterizations and properties of right Nil-pure rings, which is a proper generalization of every ideal of R is pure. And we study the regularity of right Nil-pure ring. For example: 1-Let R be a reversi...

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...

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 suc...

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 -...

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 a...

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 a...

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 singu...

ABSTRACT As a generalization of right weakly regular rings, we introduce the notion of right n-weakly regular rings, i.e. for all a  N(R), aaRaR. 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 rin...

R ‫ﺍﻟﻨﻤﻁ‬ ‫ﻤﻥ‬ ‫ﻭﻏﺎﻤﺭﺓ‬ ‫ﻤﻌﻤﻤﺔ‬ ‫ﻤﻨﻔﺭﺩﺓ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺤﻠﻘﺔ‬ ‫ﺒﺄﻨﻬﺎ‬-AP ‫ﺇﺫﺍ‬ ‫ﺃﻋﻅﻤﻲ‬ ‫ﻤﺜﺎﻟﻲ‬ ‫ﻜل‬ ‫ﻜﺎﻥ‬ ‫ﺃﻴﻤﻥ‬ ‫ﺃﺴﺎﺴﻲ‬ M ‫ﻓﻲ‬ R ‫ﻭﻟﻜل‬ b∈M ‫ﻓﺎﻥ‬ bR/bM ‫ﺍﻟﻨﻤﻁ‬ ‫ﻤﻥ‬ ‫ﻏﺎﻤﺭ‬-AP. ‫ﻫﺫﺍ‬ ‫ﻭﺨﻭﺍﺹ‬ ‫ﻤﻤﻴﺯﺍﺕ‬ ‫ﺒﺩﺭﺍﺴﺔ‬ ‫ﻗﻤﻨﺎ‬ ‫ﺍﻟﺤﻠﻘﺎﺕ‬ ‫ﻤﻥ‬ ‫ﺍﻟﺼﻨﻑ‬. ‫ﺒﺼ‬ ‫ﻋﺎﻤﺔ‬ ‫ﻭﺭﺓ‬ ، ‫ﻤﺎ‬ ‫ﻤﻥ‬ ‫ﻭﺍﻟﻐﺎﻤﺭﺓ‬ ‫ﺍﻟﻤﻌﻤﻤﺔ‬ ‫ﺍﻟﻤﻨﻔﺭﺩﺓ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﻟﻠﺤﻠﻘﺔ‬ ‫ﺍﻟﺸﺭﻭﻁ‬ ‫ﻫﻲ‬ ‫ﺍﻟﻨﻤﻁ‬-AP ‫ﻨﻴﻭﻤﺎﻥ‬ ‫...

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 r...

ABSTRACT As a generalization of regular rings, that is a ring is called n-regular if aRa a ∈ for all) (R N a ∈. 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 ∈ I, there exists b ∈ I and a positive integer n such that a n ≠ 0 and a n = ba n. In this paper, we first give and develope various properties of right N-flat rings, by which, many of the known results are extended. Also, we study the relations between such rings a...

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.

ABSTRACT A right module M is called Wnil-injective if for any 0 ≠ a ∈ N(R), there exists a positive integer n such that a n ≠ 0 and any right R-homomorphism f : a n R → 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 s...

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 fla...

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 regu...

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.

In this paper, we continue, like several other authors, to study weakly π-regular rings. In particular, we investigate some characterizations and several basic properties of these rings and the relationship between them and simple rings, strongly π-regular rings, the maximality of prime ideals in ERT rings, exchange rings and Kasch rings, respectiv...

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.

MEP) ‫ﺍﻟ‬ ‫ﻤﺜـﺎﻟﻲ‬ ‫ﺠﺯﺀ‬ ‫ﻜل‬ ‫ﻓﻴﻬﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺤﻠﻘﺎﺕ‬ ‫ﺍﻴﻤﻥ‬ ‫ﺃﻋﻅﻤﻲ‬ ‫ﺃﺴﺎﺴﻲ‬ ‫ﻨﻘﻲ‬ ‫ﻫﻭ‬ ‫ﺃﻴﺴﺭ‬ (‫ﻭﺇﻋﻁﺎﺀ‬ ‫ﻟﻬـﺎ‬ ‫ﺍﻷﺴﺎﺴﻴﺔ‬ ‫ﺍﻟﺨﻭﺍﺹ‬. ‫ﺍﻟـﺸﺭﻭﻁ‬ ‫ﺇﻋﻁـﺎﺀ‬ ‫ﻜـﺫﻟﻙ‬ ‫ﻟﻠﺤﻠﻘﺔ‬ ‫ﻭﺍﻟﻜﺎﻓﻴﺔ‬ ‫ﺍﻟﻀﺭﻭﺭﻴﺔ‬ MEP ‫ﺒﻀﻌﻑ‬ ‫ﻭﻤﻨﺘﻅﻤﺔ‬ ‫ﺒﻘﻭﺓ‬ ‫ﻤﻨﺘﻅﻤﺔ‬ ‫ﺤﻠﻘﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﻟﻜﻲ‬. ABSTRACT This paper introduces the notion of a right MEP-ring (a ring in which every maximal essential r...

Abstract 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,...

ABSRACT 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)⊆r(a) for any a∈R. 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 sa...

MRGP ‫ﺍﻟﻘﺴﻤﺔ‬ ‫ﻭﺤﻠﻘﺔ‬ ‫ﻜﺎﺵ‬ ‫،ﺤﻠﻘﺔ‬ ‫ﺒﻘﻭﺓ‬ ‫ﺍﻟﻤﻨﺘﻅﻤﺔ‬ ‫ﺍﻟﺤﻠﻘﺔ‬ ‫ﻤﻊ‬ ‫ﻭﻋﻼﻗﺘﻬﺎ‬ ‫ﺨﻭﺍﺼﻬﺎ‬ ‫ﺒﻌﺽ‬ ‫ﻋﺭﺽ‬ ‫ﻭﺘﻡ‬. ABSTRACT 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-r...

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.

PROVE THAT zero commutative strongly II- regular ring

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.