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On Herstein's Lie map conjectures, I


Authors: K. I. Beidar, M. Bresar, M. A. Chebotar and W. S. Martindale III
Journal: Trans. Amer. Math. Soc. 353 (2001), 4235-4260
MSC (1991): Primary 16W10, 16W20, 16R50
DOI: https://doi.org/10.1090/S0002-9947-01-02731-3
Published electronically: June 6, 2001
MathSciNet review: 1837230
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Abstract:

We describe surjective Lie homomorphisms from Lie ideals of skew elements of algebras with involution onto noncentral Lie ideals (factored by their centers) of skew elements of prime algebras ${\mathcal{D}}$ with involution, provided that $\operatorname{char}({\mathcal{D}})\not=2$ and ${\mathcal{D}}$ is not PI of low degree. This solves the last remaining open problem of Herstein on Lie isomorphisms module cases of PI rings of low degree. A more general problem on maps preserving any polynomial is also discussed.


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Additional Information

K. I. Beidar
Affiliation: Department of Mathematics, National Cheng-Kung University, Tainan, Taiwan
Email: beidar@mail.ncku.edu.tw

M. Bresar
Affiliation: Department of Mathematics, PF, University of Maribor, Maribor, Slovenia
Email: bresar@uni-mb.si

M. A. Chebotar
Affiliation: Department of Mechanics and Mathematics, Tula State University, Tula, Russia
Email: mchebotar@tula.net

W. S. Martindale III
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
Email: jmartind@chapline.net

DOI: https://doi.org/10.1090/S0002-9947-01-02731-3
Received by editor(s): October 6, 1999
Received by editor(s) in revised form: June 1, 2000
Published electronically: June 6, 2001
Additional Notes: The second author was partially supported by a grant from the Ministry of Science of Slovenia
Article copyright: © Copyright 2001 American Mathematical Society

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