Lie maps on alternative rings preserving idempotents

Bruno Leonardo Macedo Ferreira, Henrique Guzzo Jr., Ivan Kaygorodov Colloquium Mathematicum MSC: Primary 17A36; Secondary 17D05. DOI: 10.4064/cm8195-10-2020 Published online: 1 April 2021


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.


  • Bruno Leonardo Macedo FerreiraFederal University of Technology
    Professora Laura Pacheco Bastos Avenue, 800
    85053-510, Guarapuava, Brazil
  • Henrique Guzzo Jr.Institute of Mathematics
    University of São Paulo
    Matão Street, 1010
    05508-090, São Paulo, Brazil
  • Ivan KaygorodovFederal University of ABC
    dos Estados Avenue, 5001
    09210-580, Santo André, Brazil
    Moscow Center for Fundamental and Applied Mathematics
    Moscow, 119991, Russia

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