On Herstein’s Lie map conjectures, I

Authors:
K. I. Beidar, M. Brešar, 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 | References | Similar Articles | Additional Information

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

Article copyright:
© Copyright 2001
American Mathematical Society