Characterization of Lie multiplicative derivation on alternative rings

Abstract

In this paper we generalize the result valid for associative rings due \cite[Martindale III]{Mart} and \cite[Bre$\check{s}$ar]{bresar} to alternative rings. Let $\mathfrak{R}$ be an unital alternative ring, and $\mathfrak{D}: \mathfrak{R} \rightarrow \mathfrak{R}$ is a Lie multiplicative derivation. Then $\mathfrak{D}$ is the form $\delta + \tau$ where $\delta$ is an additive derivation of $\mathfrak{R}$ and $\tau$ is a map from $\mathfrak{R}$ into its center $\mathcal{Z}(\mathfrak{R})$, which maps commutators into the zero.