Salah Mehdi Salih

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• Professor Ph.D.
• Faculty Member at Mustansiriyah University

The purpose of this paper is to study the concept of orthogonal generalized higher symmetric reverse bi-derivation on semiprime Γ-ring. We study some lemmas and theorems of orthogonality on semiprime Γ-rings. We prove that if M is a 2-tortion free semiprime Γ-ring then D n and G n are orthogonal generalized higher symmetric reverse bi-derivations a...

The object of this article is present the orthogonally of different two concepts. We proved: Assume and are orthog gd and l(rr)c on 2-tortion free semiprime ring s.t. is commutative then and are orthog where is associated derivation of .

The volume 4, 2022 of International Journal of Mathematical Combinatorics

The definition of orthogonal generalized higher k-derivation is examined in this paper and we introduced some of its related results.

In this paper, we introduce the concepts of generalized symmetric higher bi-left (resp. right) centralizers, Jordan generalized symmetric higher bi-left (resp. right) centralizers and Jordan generalized symmetric higher triple bi-left (resp. right) centralizers on Γ-ring and we proved that every Jordan generalized symmetric higher bi-left (resp. ri...

In this paper we study the concept of orthogonal generalized higher (σ,τ)-k-derivations and present some of result obtained from orthogonality.

In this paper we introduce the concepts of generalized higher left centralizer and generalized Jordan higher left centralizer of Γ-rings M as well as we proved that every generalized Jordan higher left centralizer of certain Γ-ring M is generalized higher left centralizer of M and we prove every Jordan generalized higher left centralizer of certain...

The propos of this paper are preset the concepts of generalized (,)-reverse derivation, Jordan generalized (σ,τ)-reverse derivation and Jordan generalized triple (σ,τ)-reverse derivation on semiring. We prove that every Jordan generalized (σ,τ)-reverse derivations on semiring S with additive inverse and identity associated with Jordan (σ,τ)-reverse...

Let R , R' be two prime rings and  n , n be two higher homomorphisms of a ring R for all n  N , in the present paper we show that under certain conditions of R, every generalized Jordan (,)-higher homomorphism of a ring R into a prime ring R' is either generalized (,)-higher homomorphism or (,)-higher anti homomorphism.

Let R be a ring, in This paper we introduce the concepts of higher reverse left (resp. right) centralizer, Jordan higher reverse left (resp. right) centralizer and Jordan triple higher reverse left (resp. right) centralizer of Prime rings.

Specialists in the field of algebra have highlighted many results related to the Γ-ring, and in my research, I will present some definitions and concepts and a major theorem parallel to the results of other researchers.

In this paper we generalize the results of Md.Fazlud Hoque, and A.C.Paul and B.Zalar on generalized centralizer of completely prime -ring. We prove that every Jordan generalized centralizer of a 2-torsion free completely prime -ring M is a generalized centralizer.

In this study we introduce the concept of orthogonal generalized higher reverse left (resp.right) centralizers of -rings. The most important findings of this paper are as follows: let M be a 2-torsion free semiprime -ring, T=(T i) iN and H=(H i) iN be two generalized higher reverse left (resp.right) centralizers of M. Then T n and Hn are orthog...

In this paper, we present the concept of triple higher reverse derivation and study this concept with Jordan triple higher reverse derivation. The aim proposes paper is proving that every Jordan triple higher reverse derivation of a ring R is triple higher reverse derivation of R. 1) Introduction Through this paper R will denoted an associative rin...

In this paper we present the concept of orthogonal generalized higher reverse derivations on semiprime -ring, also we prove the following resultes if and are orthogonal such that , then the following relations hold: (1). (2). (3) and. In addition to demonstration other.

In this paper we introduce the concepts of (-reverse derivation, Jordan (σ,τ)-reverse derivation and Jordan triple (σ τ-reverse derivation on semiring. We prove that every Jordan (σ τ-reverse derivation on 2-torsion free semiring S is Jordan triple (σ τ-reverse derivation.

In this paper we present the concepts of higher-derivations and Jordan higher-derivations from Γ-near-ring M into ΓM-modules X also we prove that every Jordan higher-derivation from Γ-near-ring M into 2-torsion free ΓM-module X such that aαbβc=aβbαc for all a, b, cM and α,βΓ is Jordan triple higher-derivation of M into X. ‫المستخلص:‬ ‫المشتقات‬ ‫...

Let R be a ring, in This paper we introduce the concepts of higher reverse left (resp. right) centralizer, Jordan higher reverse left (resp. right) centralizer and Jordan triple higher reverse left (resp. right) centralizer of Prime rings.

In this study, we introduce and study the concepts of generalized ( , )-reverse derivation, Jordan generalized ( , )-reverse derivation, and Jordan generalized triple ( , )-reverse derivation from Γ-semiring S into ΓS-module X. The most important findings of this paper are as follows: If S is Γ-semiring and X is ΓS-module, then every Jordan general...

hrough this paper we define the higher triple left resp. right centralizers of a Γ-ring Ɠ, and study some properties of Jordan higher triple left resp. right centralizers of Ɠ, addition to we prove that: every Jordan higher triple left resp- right centralizer of a Γ-ring Ɠ is higher triple left resp. right centralizer f Ɠ when Ɠ is a 2-torsion free...

In this research, we will introduce generalized high Homorphics, and explain certain features of generalized high Homorphics, as well as discuss certain significant connections and distinctions..

Let R be a semiprime ring ,we prove the following main result : Let R be a 2-torsion free semiprime ring , t=(t i) iN and h=(h i) iN be two higher reverse left (resp.right) centralizers of R .Then t n and h n are orthogonal if and only if t n (x) h n (y) + h n (x) t n (y) = 0 , for all x , y  R and n N .

In this paper, we introduce the concepts of higher reverse left (resp.right) centralizer, Jordan higher reverse left (resp. right) centralizer, and Jordan triple higher reverse left (resp. right) centralizer of -rings. We prove that every Jordan higher reverse left (resp. right) centralizer of a 2-torsion free prime -ring M is a higher reverse le...

The main object of this paper is prove that: Let R be a 2-torsion free semiprime ring ,T=(T i) iN and H=(H i) iN be two generalized higher reverse left (resp. right) centralizers associated with the higher reverse left (resp. right) centralizers t=(t i) iN and h=(h i) iN resp. of R , where T n and H n are commuting. Then T n and H n are orthogo...

In this paper we introduce the concept of orthogonal higher reverse left (resp. right) centralizers on semiprime -ring and we prove the following main result: Let M be a 2-torsion free semiprime -ring, t=(t i) iN and h=(h i) iN be two higher reverse left (resp.right) centralizers of M,suppose that t n (x)  t n (x) = h n (x)  h n (x), for all...

The concepts of generalized higher derivations, Jordan generalized higher derivations, and Jordan generalized triple higher derivations on Γ-ring M into ΓM-modules X are presented. We prove that every Jordan generalized higher derivation of Γ-ring M into 2-torsion free ΓM-module X, such that aαbβc=aβbαc, for all a, b, c M and α,βΓ, is Jordan gene...

Orthogonal Symmetric Higher bi-Derivations í µí²í µí² í µí°’í µí°ží µí°¦í µí°¢í µí°©í µí°«í µí°¢í µí°¦í µí°ž Г-Rings í µí±ºí µí²‚í µí²í µí²‚í µí²‰ í µí±´í µí²†í µí²‰í µí²í µí²Š í µí±ºí µí²‚í µí²í µí²Ší µí²‰ í µí±ºí µí²‚í µí²Ží µí²‚í µí²‰ í µí±±í µí²‚í µí²ƒí µí²†í µí²“ í µí±ºí µí²‰í µí²‚í µí²Œí µí²†í µí²“ Abstract: Let M is a Г-ring. In this pap...

In this paper, the concept of Jordan triple higher-homomorphisms on prime rings is introduced. A result of Herstein is extended on this concept from the ring into the prime ring. We prove that every Jordan triple higher-homomorphism of ring into prime ring is either triple higher-homomorphism or triple higher-anti-homomorphism of into .

In this paper, we introduce the concepts of higher reverse left (resp.right) centralizer, Jordan higher reverse left (resp. right) centralizer, and Jordan triple higher reverse left (resp. right) centralizer of -rings. We prove that every Jordan higher reverse left (resp. right) centralizer of a 2-torsion free prime -ring M is a higher reverse le...

In this paper we define cr-metric space, cr-compact space, o-connected space and cr-metrizable space also rve present the relations among them and metric space, compact space, connected space and metrizable respectively. :i-^a)6,11 ,r-Loi-In this 'rvork rve investigate cr-metric space also we give the results about cr-metric spaces. In $2 rve defin...

In this paper lve define the semi homotopy, strongly semi homotopy, and we study the relation among them and homotopy, cr-homotopy and strongly q-homotopy, also we define semi isotopy and strongly semi isotopy. and we study the relation among them and isotopy, cr-isotopy and strongly isotopy.

Let í µí± be a ring and í µí¼Ž, í µí¼ be an endomorphisms of í µí±, in this paper we will present and study the concepts of higher (í µí¼Ž, í µí¼)-centralizer, Jordan higher(í µí¼Ž, í µí¼)-centralizer and Jordan triple higher (í µí¼Ž, í µí¼)-centralizer and their generalization on the ring. The main results are prove that every Jordan higher (í...

Let M be a semiprime -ring satisfying a certain assumption. Then we prove that every Jordan left higher k-centralizer on M is a left higher k-centralizer on M. We also prove that every Jordan higher k-centralizer of a 2-torsion free semiprime -ring M satisfying a certain assumption is a higher k-centralizer.

In this paper some results concerning to right reverse derivation on prime -rings are presented if M be a prime -ring with non-zero right reverse derivation d and U be the ideal of M, then M is commutative. Mathematics Subject Classification: 16A70, 16N60, 16W25

The aim object of this paper is present the definitions of drivation, Jordan derivation and Jordan triple derivation on-M-module also we prove that every Jordan derivation of "-ring M into 2-torsion prime -module X is a derivation of M into X. Mathematics Subject Classification: 16W30, 16W25.

In this study , we define the concepts of a generalized higher bi-derivation , Jordan generalized higher bi-derivation and Jordan triple generalized higher bi-derivation on rings and show that a Jordan generalized higher bi-derivation on 2-torsion free prime ring is a generalized higher bi-derivation .

The aim object of this paper is present the definitions of generalized drivation, Jordan generalized derivation and Jordan generalized triple derivation of -rings into -module also we prove that every Jordan generalized derivation of Γ-ring M into 2-torsion prime -module X is a derivation of M into X. Mathematics Subject Classification: 16W30,...

In this study , we define the concepts of a generalized higher bi-derivation , Jordan generalized higher bi-derivation and Jordan triple generalized higher bi-derivation on Г-rings and show that a Jordan generalized higher bi-derivation on 2-torsion free prime Г-ring is a generalized higher bi-derivation. 1.Introduction Let M and Г be two additive...

Let R , R' be two prime rings and  n , n be two higher homomorphisms of a ring R for all n  N , in the present paper we show that under certain conditions of R, every generalized Jordan (,)-higher homomorphism of a ring R into a prime ring R' is either generalized (,)-higher homomorphism or (,)-higher anti homomorphism.

In this paper, we develop some important results relating to the concepts of triple higher derivation and Jordan triple higher derivation on ring R. We show that under certain conditions on R, every Jordan triple higher derivation on R is triple higher derivation on R.

This study introduces the concepts of generalized higher reverse left (respectively right) centralizer , Jordan generalized higher reverse left (respectively right) centralizer and Jordan triple generalized higher reverse left (respectively right) centralizer of Gamma-rings. In this paper we prove the following main results. Every Jordan generalize...

This study introduces the concepts of generalized higher reverse left (respectively right) centralizer , Jordan generalized higher reverse left (respectively right) centralizer and Jordan triple generalized higher reverse left (respectively right) centralizer of Gamma-rings. In this paper we prove the following main results. Every Jordan generalize...

In this study we introduced the concepts of generalized higher reverse left (accordingly, right) centralizer, and Jordan generalized higher reverse left (accordingly, right) centralizer of rings. The definition of Jordan triple generalized higher reverse left (accordingly, right) centralizer was deduced. The most important findings of this paper ar...

Let í µí±€ be a 2-torsion free prime í µí»¤-ring. Then we prove that every Jordanhigher left (í µí¼Ž, í µí¼)-centralizer on í µí±€ is higher left (í µí¼Ž, í µí¼)-centralizer on M. We also prove that with certain conditions every Jordan higher left (í µí¼Ž, í µí¼)-centralizer on í µí±€ is a Jordan triple higherleft(í µí¼Ž, í µí¼)-centralizer of...

In this paper we introduce and study the concept of orthogonal generalized higher reveres derivations on semi prime rings , also we prove the following theorem if í µí°· í µí±› and í µí°º í µí±› are orthogonal, then the following relations hold: (1) í µí°· í µí±› í µí±¥ í µí°º í µí±› í µí±¦ = í µí°º í µí±› í µí±¥ í µí°· í µí±› í µí±¦ = 0 (2) í µí±‘...

In this paper a Г-ring M is presented. We will study the concept of orthogonal generalized symmetric higher bi-derivations on Г-ring. We prove that if M is a 2-torsion free semiprime Г-ring , and are orthogonal generalized symmetric higher bi-derivations associated with symmetric higher bi-derivations respectively for all n ϵN. Then the following r...

International Journal of Advanced Scientific and Technical Research http://www.rspublication.com/ijst/index.html Issue 6 volume 4, July –August 2016 ISSN 2249-9954 Abstract: The aim object of our work is present the concept of generalized triple higher reverse derivation and prove that every Jordan generalized triple higher reverse derivation of...

In this paper we study the commutativity of prime rings satisfying certain identities involving higher left centralizer on it. Math. Classification QAISO-27205 Key words: prime rings , higher left centralizer 1.Introduction : Throughout this paper í µí± is denote to an associative ring and it is center will denoted by í µí±(í µí±) which equal to t...

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) http://iiste.org/Journals/index.php/MTM/article/view/25407/26326 In this paper, we develop some important results relating to the concepts of triple higher derivation and Jordan triple higher derivation on ring R. We show that under certain conditions on R, every Jorda...

Let M be a -ring and ,  be two endomorphisms of M. In this paper, some result on the centralizing of (,)-derivations on a subset S of a prime -ring M. Also we study the commutativity of M by using the concepts centralizing and commuting of a (,)-derivations of M.

Let R , R' be two prime rings and  n , n be two higher homomorphisms of a ring R for all n  N , in the present paper we show that under certain conditions of R, every generalized Jordan (,)-higher homomorphism of a ring R into a prime ring R' is either generalized (,)-higher homomorphism or (,)-higher anti homomorphism.

In this paper we study the commutativity of prime-rings satisfying certain identities involving higher left centralizer on it .

In this paper we study the commutativity of prime-rings satisfying certain identities involving higher left centralizer on it .

In this paper, we study the concepts of generalized reverse derivation, Jordan generalized reverse derivation and Jordan generalized triple reverse derivation on -ring M. The aim of this paper is to prove that every Jordan generalized reverse derivation of -ring M is generalized reverse derivation of M.

Let M and M' be two prime -rings and  n , n be two higher homomorphism of a -ring M, for all n  N in the present paper we show that under certain conditions of M, every Jordan (,)-higher homomorphism of a -Ring M into a prime  '-Ring M 'is either (,)-higher homomorphism or (,)-anti-higher homomorphism.

Let M and M' be two prime -ring and  n , n be two higher homomorphism of a -ring M, for all n  N in the present paper we show that under certain conditions of M, every generalized Jordan (σ, τ)-higher homomorphism of a -Ring M into a prime -Ring M 'is either generalized (σ, τ)-higher homomorphism or (σ, τ)-higher anti-homomorphism. Mathemati...

Let M and M' be two prime -ring and  n , n be two higher homomorphism of a -ring M, for all n  N in the present paper we show that under certain conditions of M, every generalized Jordan (σ, τ)-higher homomorphism of a -Ring M into a prime -Ring M 'is either generalized (σ, τ)-higher homomorphism or (σ, τ)-higher anti-homomorphism. Mathemati...

Let M be Γ-ring and X be ΓM-module, Bresar and Vukman studied orthogonal derivations on semiprime rings. Ashraf and Jamal defined the orthogonal derivations on -rings M. This research defines and studies the concepts of orthogonal derivation and orthogonal generalized derivations on ΓM-Module X and introduces the relation between the products of g...