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A NOTE ON MULTIPLICATIVE (GENERALIZED)-DERIVATION IN SEMIPRIME RINGS
Nadeem Ur Rehman,Motoshi Hongan 한국전산응용수학회 2018 Journal of applied mathematics & informatics Vol.36 No.1
In this article we study two Multiplicative (generalized)- derivations $\mathcal{G}$ and $\mathcal{H}$ that satisfying certain conditions in semiprime rings and tried to find out some information about the associated maps. Moreover, an example is given to demonstrate that the semiprimeness imposed on the hypothesis of the various results is essential.
A NOTE ON MULTIPLICATIVE (GENERALIZED)-DERIVATION IN SEMIPRIME RINGS
REHMAN, NADEEM UR,HONGAN, MOTOSHI The Korean Society for Computational and Applied M 2018 Journal of applied mathematics & informatics Vol.36 No.1
In this article we study two Multiplicative (generalized)- derivations ${\mathcal{G}}$ and ${\mathcal{H}}$ that satisfying certain conditions in semiprime rings and tried to find out some information about the associated maps. Moreover, an example is given to demonstrate that the semiprimeness imposed on the hypothesis of the various results is essential.
Identities in a Prime Ideal of a Ring Involving Generalized Derivations
ur Rehman, Nadeem,Ali Alnoghashi, Hafedh Mohsen,Boua, Abdelkarim Department of Mathematics 2021 Kyungpook mathematical journal Vol.61 No.4
In this paper, we will study the structure of the quotient ring R/P of an arbitrary ring R by a prime ideal P. We do so using differential identities involving generalized derivations of R. We enrich our results with examples that show the necessity of their assumptions.
Derivations with Power Values on Lie Ideals in Rings and Banach Algebras
Rehman, Nadeem ur,Muthana, Najat Mohammed,Raza, Mohd Arif Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.2
Let R be a 2-torsion free prime ring with center Z, U be the Utumi quotient ring, Q be the Martindale quotient ring of R, d be a derivation of R and L be a Lie ideal of R. If $d(uv)^n=d(u)^md(v)^l$ or $d(uv)^n=d(v)^ld(u)^m$ for all $u,v{\in}L$, where m, n, l are xed positive integers, then $L{\subseteq}Z$. We also examine the case when R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non-commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.
A REMARK ON GENERALIZED DERIVATIONS IN RINGS AND ALGEBRAS
Nadeem Ur Rehman 한국수학교육학회 2018 純粹 및 應用數學 Vol.25 No.3
In the present paper, we investigate the action of generalized derivation G associated with a derivation g in a (semi-) prime ring R satisfying (G([x; y]) - [G(x); y])n = 0 for all x, y ∈ I, a nonzero ideal of R, where n is a fixed positive integer. Moreover, we also examine the above identity in Banach algebras.
A Note on Skew-commuting Automorphisms in Prime Rings
ur Rehman, Nadeem,Bano, Tarannum Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.1
Let R be a prime ring with center Z, I a nonzero ideal of R, and ${\sigma}$ a non-trivial automorphism of R such that $\{(x{\circ}y)^{\sigma}-(x{\circ}y)\}^n{\in}Z$ for all $x,y{\in}I$. If either char(R) > n or char (R) = 0, then R satisfies $s_4$, the standard identity in 4 variables.
Annihilating Conditions of Generalized Skew Derivations on Lie Ideals
Nadeem Ur Rehman,Sajad Ahmad Pary,Junaid Nisar 경북대학교 자연과학대학 수학과 2023 Kyungpook mathematical journal Vol.63 No.3
Let A be a prime ring of char(A)̸= 2, L a non-central Lie ideal of A, F a generalized skew derivation of A and p ∈ A, a nonzero fixed element. If pF(η)η ∈ C for any η ∈ L , then A satisfies S4.
NOTES ON SYMMETRIC SKEW n-DERIVATION IN RINGS
Koc, Emine,Rehman, Nadeem ur Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.4
Let R be a prime ring (or semiprime ring) with center Z(R), I a nonzero ideal of R, T an automorphism of $R,S:R^n{\rightarrow}R$ be a symmetric skew n-derivation associated with the automorphism T and ${\Delta}$ is the trace of S. In this paper, we shall prove that S($x_1,{\ldots},x_n$) = 0 for all $x_1,{\ldots},x_n{\in}R$ if any one of the following holds: i) ${\Delta}(x)=0$, ii) [${\Delta}(x),T(x)]=0$ for all $x{\in}I$. Moreover, we prove that if $[{\Delta}(x),T(x)]{\in}Z(R)$ for all $x{\in}I$, then R is a commutative ring.
Identities with additive mappings in semiprime rings
Ajda Fosner,Nadeem Ur Rehman 대한수학회 2014 대한수학회보 Vol.51 No.1
The aim of this paper is to prove the next result. Let n > 1 be an integer and let R be a n!-torsion free semiprime ring. Suppose that f : R → R is an additive mapping satisfying the relation [f(x), xn] = 0 for all x ∈ R. Then f is commuting on R.
REMARKS ON GENERALIZED JORDAN (α, β)<sup>*</sup>-DERIVATIONS OF SEMIPRIME RINGS WITH INVOLUTION
Hongan, Motoshi,Rehman, Nadeem ur Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.1
Let R be an associative ring with involution * and ${\alpha},{\beta}:R{\rightarrow}R$ ring homomorphisms. An additive mapping $d:R{\rightarrow}R$ is called an $({\alpha},{\beta})^*$-derivation of R if $d(xy)=d(x){\alpha}(y^*)+{\beta}(x)d(y)$ is fulfilled for any $x,y{\in}R$, and an additive mapping $F:R{\rightarrow}R$ is called a generalized $({\alpha},{\beta})^*$-derivation of R associated with an $({\alpha},{\beta})^*$-derivation d if $F(xy)=F(x){\alpha}(y^*)+{\beta}(x)d(y)$ is fulfilled for all $x,y{\in}R$. In this note, we intend to generalize a theorem of Vukman [12], and a theorem of Daif and El-Sayiad [6], moreover, we generalize a theorem of Ali et al. [4] and a theorem of Huang and Koc [9] related to generalized Jordan triple $({\alpha},{\beta})^*$-derivations.