Skew $N$-Derivations on Semiprime Rings

Abstract

For a ring $R$ with an automorphism $\sigma$, an $n$-additive mapping $\Delta:R\times R\times... \times R \rightarrow R$ is called a skew $n$-derivation with respect to $\sigma$ if it is always a $\sigma$-derivation of $R$ for each argument. Namely, it is always a $\sigma$-derivation of $R$ for the argument being left once $n-1$ arguments are fixed by $n-1$ elements in $R$. In this short note, starting 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 $R$.