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