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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On Herstein’s Lie map conjectures, I
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by K. I. Beidar, M. Brešar, M. A. Chebotar and W. S. Martindale III PDF
Trans. Amer. Math. Soc. 353 (2001), 4235-4260 Request permission

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. Brešar
  • 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
  • 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
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1837230