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Commutators and anti-commutators having automorphisms on Lie ideals in prime rings
Mohd Arif Raza,Hussain Alhazmi 강원경기수학회 2020 한국수학논문집 Vol.28 No.3
In this manuscript, we discuss the relationship between prime rings and automorphisms satisfying differential identities involving commutators and anti-commutators on Lie ideals. In addition, we provide an example which shows that we cannot expect the same conclusion in case of semiprime rings.
ON AUTOMORPHISMS IN PRIME RINGS WITH APPLICATIONS
Raza, Mohd Arif Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.4
The notions of skew-commuting/commuting/semi-commuting/skew-centralizing/semi-centralizing mappings play an important role in ring theory. ${\mathfrak{C}}^*$-algebras with these properties have been studied considerably less and the existing results are motivating the researchers. This article elaborates the structure of prime rings and ${\mathfrak{C}}^*$-algebras satisfying certain functional identities involving automorphisms.
A NOTE ON GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS
RAZA, MOHD ARIF,REHMAN, NADEEM UR,GOTMARE, A.R. The Korean Society for Computational and Applied M 2021 Journal of applied mathematics & informatics Vol.39 No.1
Let R be a prime ring, Qr be the right Martindale quotient ring and C be the extended centroid of R. If be a nonzero generalized skew derivation of R and f(x1, x2, ⋯, xn) be a multilinear polynomial over C such that ((f(x1, x2, ⋯, xn)) - f(x1, x2, ⋯, xn)) ∈ C for all x1, x2, ⋯, xn ∈ R, then either f(x1, x2, ⋯, xn) is central valued on R or R satisfies the standard identity s4(x1, x2, x3, x4).
Non-linear product $LM^\ast -ML^\ast $ on prime $\ast-$algebras
Mohd Arif Raza,Tahani Al-Sobhi 강원경기수학회 2023 한국수학논문집 Vol.31 No.3
In this paper, we explore the additivity of the map $\Omega :\mathrsfso{A}\rightarrow \mathrsfso{A}$ that satisfies $$\Omega\left( [\mathrsfso{L},\mathrsfso{M}]_{*} \right)=[\Omega\left( \mathrsfso{M}\right),\mathrsfso{L}] _{*} + [\mathrsfso{M}, \Omega\left( \mathrsfso{L}\right)]_{*},$$ where $[\mathrsfso{L}, \mathrsfso{M}] _{*}= \mathrsfso{L}\mathrsfso{M}^\ast -\mathrsfso{M} \mathrsfso{L}^\ast$, for all $\mathrsfso{L},\mathrsfso{M} \in\mathcal {\mathrsfso{A} }$, a prime $\ast-$algebra with unit $\mathrsfso{I}$. Additionally we show that if ${\Omega}(\alpha \mathrsfso{I})$ is self-adjoint operator for $ \alpha \in\{1, i\} $, then $\Omega=0$.
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.
Commutativity of multiplicative $b$-generalized derivations of prime rings
Muzibur Rahman Mozumder,Wasim Ahmed,Mohd Arif Raza,Adnan Abbasi 강원경기수학회 2023 한국수학논문집 Vol.31 No.1
Consider $\mathscr{R}$ to be an associative prime ring and $\mathscr{K}$ to be a nonzero dense ideal of $\mathscr{R}$. A mapping (need not be additive) $\mathscr{F} : \mathscr{R} \rightarrow \mathscr{Q} _{mr}$ associated with derivation $d : \mathscr{R} \rightarrow \mathscr{R}$ is called a multiplicative $b$-generalized derivation if $\mathscr{F} (\alpha \delta ) = \mathscr{F} (\alpha )\delta + b\alpha d(\delta )$ holds for all $\alpha ,\delta \in \mathscr{R}$ and for any fixed $(0 \neq) b \in \mathscr{Q}_s \subseteq \mathscr{Q}_{mr}$. In this manuscript, we study the commutativity of prime rings when the map $b$-generalized derivation satisfies the strong commutativity preserving condition and moreover, we investigate the commutativity of prime rings that admit multiplicative $b$-generalized derivation, which improves many results in the literature.