Skew $n$-derivations on semiprime rings
Bull. Korean Math. Soc. 2013 Vol. 50, No. 6, 1863-1871
Printed November 1, 2013
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
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