Lie maps on alternative rings

Abstract

Let ${\mathfrak {R}}\,$ be an alternative ring containing a nontrivial idempotent and ${\mathfrak {R}}\, '$ be another alternative ring. Suppose that a bijective mapping $\varphi : {\mathfrak {R}}\, \rightarrow {\mathfrak {R}}\, '$ is a Lie multiplicative mapping and ${\mathfrak {D}}\,$ is a Lie triple derivable multiplicative mapping from ${\mathfrak {R}}\,$ into ${\mathfrak {R}}\,$. Under a mild condition on ${\mathfrak {R}}\,$, we prove that $\varphi$ and ${\mathfrak {D}}\,$ are almost additive, that is, $\varphi (a + b) - \varphi (a) - \varphi (b) \in \mathcal {Z}({\mathfrak {R}}\, ')$ and ${\mathfrak {D}}\, (a+b) - {\mathfrak {D}}\, (a) - {\mathfrak {D}}\, (b) \in \mathcal {Z}({\mathfrak {R}}\, )$ for all $a,b \in {\mathfrak {R}}\,$, where $\mathcal {Z}({\mathfrak {R}}\, ')$ is the commutative centre of ${\mathfrak {R}}\, '$ and $\mathcal {Z}({\mathfrak {R}}\, )$ is the commutative centre of ${\mathfrak {R}}\,$. As applications, we show that every Lie multiplicative bijective mapping and Lie triple derivable multiplicative mapping on prime alternative rings are almost additive.