IJPAM: Volume 115, No. 2 (2017)

Title

ON $(m,n)$-JORDAN $*$-DERIVATIONS
AND RELATED MAPPINGS IN RINGS WITH INVOLUTION

Authors

Husain Alhazmi$^1$, Shakir Ali$^2$, Mohammad Salahuddin Khan$^3$
$^{1,2}$Department of Mathematics
Faculty of Science
King Abdulaziz University
Jeddah, 21589, SAUDI ARABIA
$^3$Department of Applied Mathematics
Aligarh Muslim University
Aligarh, 202002, INDIA

Abstract

The aim of the present paper is to introduce the notions of $(m,n)~*$-derivations and $(m,n)$-Jordan $*$-derivations, and to prove some related results in rings with involution. In particular, we prove that if $(m+n+1)!$-torsion free $*$-ring $R$ admits an additive mapping $d:R\to R$ such that $d(x^{m+n+1})=(m+n+1)x^nd(x)(x^*)^m$ for all $x\in R,$ then $d$ is an $(m,n)$-Jordan $*$-derivation on $R,$ where $m\geq 0,~n\geq0$ with $m+n\neq0$ are some fixed integers. Further, we prove that if $R$ is an $(n+1)!$-torsion free, then an additive mapping $T$ satisfies $T(x^{n+1})=T(x)(x^*)^{n}$ for all $x\in R$ must be a Jordan left $*$-centralizer. As an application, Jordan left $*$-centralizers are characterized.

History

Received: November 16, 2017
Revised: May 1, 2017
Published: July 14, 2017

AMS Classification, Key Words

AMS Subject Classification: 16W10, 16W25, 16N60
Key Words and Phrases: involution, torsion free ring, $*$-ring, semiprime $*$-ring, $(m,n)$-Jordan $*$-derivation, Jordan left $*$-centralizer

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How to Cite?

DOI: 10.12732/ijpam.v115i2.19 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 115
Issue: 2
Pages: 445 - 453


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