Characterization of Lie multiplicative derivation on alternative rings

TL;DR

This paper extends the understanding of Lie multiplicative derivations from associative to alternative rings, showing such derivations decompose into additive derivations plus central maps that vanish on commutators.

Contribution

It generalizes known results for associative rings to the broader class of alternative rings, characterizing Lie multiplicative derivations as sums of derivations and central maps.

Findings

Lie multiplicative derivations decompose into additive derivations and central maps.

The central maps vanish on commutators.

The result applies to unital 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.