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Vladikavkazskii Matematicheskii Zhurnal, 2021, Volume 23, Number 2, Pages 70–77
DOI: https://doi.org/10.46698/d4945-5026-4001-v
(Mi vmj765)
 

A note on semiderivations in prime rings and $\mathscr{C}*$-algebras

M. A. Razaa, N. Rehmanb

a Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia
b Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India
References:
Abstract: Let $\mathscr{R}$ be a prime ring with the extended centroid $\mathscr{C}$ and the Matrindale quotient ring $\mathscr{Q}$. An additive mapping $\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}$ is called a semiderivation associated with a mapping $\mathscr{G}: \mathscr{R}\rightarrow \mathscr{R}$, whenever $ \mathscr{F}(xy)=\mathscr{F}(x)\mathscr{G}(y)+x\mathscr{F}(y)= \mathscr{F}(x)y+ \mathscr{G}(x)\mathscr{F}(y) $ and $ \mathscr{F}(\mathscr{G}(x))= \mathscr{G}(\mathscr{F}(x))$ holds for all $x, y \in \mathscr{R}$. In this manuscript, we investigate and describe the structure of a prime ring $\mathscr{R}$ which satisfies $\mathscr{F}(x^m\circ y^n)\in \mathscr{Z(R)}$ for all $x, y \in \mathscr{R}$, where $m,n \in \mathbb{Z}^+$ and $\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}$ is a semiderivation with an automorphism $\xi$ of $\mathscr{R}$. Further, as an application of our ring theoretic results, we discussed the nature of $\mathscr{C}^*$-algebras. To be more specific, we obtain for any primitive $\mathscr{C}^*$-algebra $\mathscr{A}$. If an anti-automorphism $ \zeta: \mathscr{A} \to \mathscr{A}$ satisfies the relation $(x^n)^\zeta+x^{n*}\in \mathscr{Z}(\mathscr{A})$ for every ${x,y}\in \mathscr{A},$ then $\mathscr{A}$ is $\mathscr{C}^{*}-\mathscr{W}_{4}$-algebra, i. e., $\mathscr{A}$ satisfies the standard identity $\mathscr{W}_4(a_1,a_2,a_3,a_4)=0$ for all $a_1,a_2,a_3,a_4\in \mathscr{A}$.
Key words: prime ring, automorphism, semiderivation.
Funding agency Grant number
NBHM 02011/16/2020 NBHM (R.P.) R & D II/ 7786
For the second author, this research is supported by the National Board of Higher Mathematics (NBHM), India, Grant № 02011/16/2020 NBHM (R.P.) R & D II/ 7786.
Document Type: Article
UDC: 512.55
MSC: 16W25, 16N60
Language: English
Citation: M. A. Raza, N. Rehman, “A note on semiderivations in prime rings and $\mathscr{C}*$-algebras”, Vladikavkaz. Mat. Zh., 23:2 (2021), 70–77
Citation in format AMSBIB
\Bibitem{RazReh21}
\by M.~A.~Raza, N.~Rehman
\paper A note on semiderivations in prime rings and $\mathscr{C}*$-algebras
\jour Vladikavkaz. Mat. Zh.
\yr 2021
\vol 23
\issue 2
\pages 70--77
\mathnet{http://mi.mathnet.ru/vmj765}
\crossref{https://doi.org/10.46698/d4945-5026-4001-v}
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