Lie maps on alternative rings preserving idempotents

Abstract

Let $\Re$ and ${\Re }^{\prime }$ unital $2$,$3$-torsion free alternative rings and $\phi :\Re \to {\Re }^{\prime }$ be a surjective Lie multiplicative map that preserves idempotents. Assume that $\Re$ has a nontrivial idempotents. Under certain assumptions on $\Re$, we prove that $\phi$ is of the form $\psi +\tau$, where $\psi$ is either an isomorphism or the negative of an anti-isomorphism of $\Re$ onto ${\Re }^{\prime }$ and $\tau$ is an additive mapping of $\Re$ into the centre of ${\Re }^{\prime }$ which maps commutators into zero.

arXiv admin note: text overlap with arXiv:2002.00304, arXiv:1802.04316