Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2018; 33(4): 1113-1121

Online first article June 28, 2018      Printed October 31, 2018

https://doi.org/10.4134/CKMS.c170454

Copyright © The Korean Mathematical Society.

Notes on symmetric skew $n$-derivation in rings

Emine Ko\c{c}, Nadeem ur Rehman

Cumhuriyet University, Aligarh Muslim University

Abstract

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.

Keywords: prime ring, semiprime ring, symmetric skew $n$-derivation, centralizing mapping, commuting mapping

MSC numbers: 16W20, 16W25, 16N60