Lie maps on alternative rings preserving idempotents
Volume 166 / 2021
Colloquium Mathematicum 166 (2021), 227-238
MSC: Primary 17A36; Secondary 17D05.
DOI: 10.4064/cm8195-10-2020
Published online: 1 April 2021
Abstract
Let $\mathfrak R$ and $\mathfrak R’$ be unital $2$,$3$-torsion free alternative rings and $\varphi : \mathfrak R \rightarrow \mathfrak R’$ be a surjective Lie multiplicative map that preserves idempotents. Assume that $\mathfrak R$ has a nontrivial idempotent. Under certain assumptions on $\mathfrak R$, we prove that $\varphi $ is of the form $\psi + \tau $, where $\psi $ is either an isomorphism or the negative of an anti-isomorphism of $\mathfrak R$ onto $\mathfrak R’$ and $\tau $ is an additive mapping of $\mathfrak R$ into the centre of $\mathfrak R’$ which maps commutators to zero.
