Radwan Mohammed Al-omary

RESEARCH, TEACHING, or OTHER INTERESTS

Mathematics, Algebra and Number Theory, Algebra and Number Theory

Scopus Publications

ON PRIME RINGS WITH INVOLUTION AND GENERALIZED DERIVATIONS
Radwan Al-omary and S. Khalid Nauman
University of Zielona Góra, Poland



COMMUTATIVITY OF PRIME RINGS WITH GENERALIZED (α, β)− REVERSE DERIVATIONS SATISFYING CERTAIN IDENTITIES
Radwan Mohammed Al-Omary
Universitatea Transilvania Brasov
In this note we investigated some conditions related to the commutativity of a prime ring R that satisfies certain identities and possesses a generalized (α, β)-reverse derivation. A few examples and counterexamples are also studied.


Generalized Derivations on Prime Rings Satisfying Certain Identities
Radwan M. Al-omary, S. Nauman and Khalid Nauman

Let R be a ring with characteristic different from 2. An additive mapping F : R → R is called a generalized derivation on R if there exists a derivation d : R → R such that F (xy) = F (x)y + xd(y) holds for all x, y ∈ R. In the present paper, we show that if R is a prime ring satisfying certain identities involving a generalized derivation F associated with a derivation d, then R becomes commutative and in some cases d comes out to be zero (i.e., F becomes a left centralizer). We provide some counter examples to justify that the restrictions imposed in the hypotheses of our theorems are not superfluous.


On (α, β) -derivations in d-algebras
Radwan Mohammed Al-Omary
Springer Science and Business Media LLC



A Note on Multiplicative (Generalized) (α, β)-Derivations in Prime Rings
Nadeem ur Rehman, Radwan M. Al-omary, and Najat Mohammed Muthana
Walter de Gruyter GmbH
Abstract Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.


Centralizing mappings, Morita context and generalized (α, β)-derivations
Nadeem ur Rehman, Motoshi Hongan, and Radwan M. Al-Omary
Informa UK Limited



Lie ideals and generalized (α, β)-derivations of *-prime rings
Nadeem ur Rehman, Radwan Mohammed AL-Omary, and Shuliang Huang
Springer Science and Business Media LLC



Lie ideals, Morita context and Generalized (α, β)-Derivations
S. Khalid NAUMAN, Nadeem ur REHMAN, and R.M. AL-OMARY
Elsevier BV



Morita context and generalized (α, β)-derivations
Nadeem Ur Rehman, Radwan Mohammed AL-Omary, and Mohammed M. Al-Shomrani
Sociedade Paranaense de Matematica
Let $R$ and $S$ be rings of a semi-projective Morita context, and $\\alpha, \\beta$ be automorphisms of $R$. An additive mapping $F$: $R\\to R$ is called a generalized $(\\alpha,\\beta)$-derivation on $R$ if there exists an $(\\alpha,\\beta)$-derivation $d$: $R\\to R$ such that $F(xy)=F(x)\\alpha(y)+\\beta(x)d(y)$ holds for all $x,y \\in R$. For any $x,y \\in R$, set $[x, y]_{\\alpha, \\beta} = x \\alpha(y) - \\beta(y) x$ and $(x \\circ y)_{\\alpha, \\beta} = x \\alpha(y) + \\beta(y) x$. In the present paper, we shall show that if the ring $S$ is reduced then it is a commutative, in a compatible way with the ring $R$ . Also, we obtain some results on bialgebras via Cauchy modules.


On commutativity of 2-torsion free *-prime rings with generalized derivations




Lie Ideals and Jordan Triple Derivations in Rings
Motoshi Hongan, Nadeem ur Rehman, and Radwan Mohammed Al-Omary
European Mathematical Society - EMS - Publishing House GmbH
. In this paper we prove that on a 2 -torsion free semiprime ring R every Jordan triple ( (cid:11);(cid:12) ) -derivation (resp. generalized Jordan triple ( (cid:11);(cid:12) ) derivation) on Lie ideal L is an ( (cid:11);(cid:12) ) -derivation on L (resp. generalized ( (cid:11);(cid:12) ) - derivation on L )


On lie structure of prime rings with generalized (α, β)-derivations
Claus Haetinger, Nadeem Ur Rehman, and Radwan Mohammed AL-Omary
Sociedade Paranaense de Matematica
Let R be a ring and , be automorphisms of R. An additive mapping F: R → R is called a generalized (\\alpha,\\beta )-derivation on R if there exists an (\\alpha,\\beta )- derivation d: R → R such that F(xy) = F(x)\\alpha(y) + \\beta (x)d(y) holds for all x, y ∈ R. For any x, y ∈ R, set [x, y]_{\\alpha,\\beta} = x\\alpha(y) − \\beta (y)x and (x o y)_{\\alpha,\\beta} = x\\alpha(y) + \\beta (y)x. In the present paper, we shall discuss the commutativity of a prime ring R admitting generalized (\\alpha,\\beta )-derivations F and G satisfying any one of the following properties: (i) F([x, y]) = (xoy)_{\\alpha,\\beta} , (ii) F(xoy) = [x, y]_{\\alpha,\\beta}, (iii) [F(x), y]_{\\alpha,\\beta} = (F(x)o y)_{\\alpha,\\beta} , (iv) F([x, y]) = [F(x), y]_{\\alpha,\\beta} , (v) F(xoy) = (F(x) o y)_{\\alpha,\\beta} , (vi) F([x, y] =[\\alpha(x),G(y)] and (vii) F(xoy) = (\\alpha(x)o G(y)) for all x, y in some appropriate subset of R. Finally, obtain some results on semi-projective Morita context with generalized (\\alpha,\\beta )-derivations.