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.