Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2013; 50(6): 1863-1871

Printed November 1, 2013

https://doi.org/10.4134/BKMS.2013.50.6.1863

Copyright © The Korean Mathematical Society.

Skew $n$-derivations on semiprime rings

Xiaowei Xu, Yang Liu, and Wei Zhang

Jilin University, Jilin University, Jilin University

Abstract

For a ring $R$ with an automorphism $\sigma$, an $n$-additive mapping $\Delta:R\times R\times\cdots \times R \rightarrow R$ is called a skew $n$-derivation with respect to $\sigma$ if it is always a $\sigma$-derivation of $R$ for each argument. Namely, if $n-1$ of the arguments are fixed, then $\Delta$ is a $\sigma$-derivation on the remaining argument. In this short note, from Bre\v{s}ar Theorems, we prove that a skew $n$-derivation ($n\geq 3$) on a semiprime ring $R$ must map into the center of $R$.

Keywords: prime ring, semiprime ring, biderivation, $n$-derivation, skew $n$-derivation

MSC numbers: 16W25, 16N60