On Herstein’s Lie map conjectures, I

Beidar, K.; Brešar, M.; Chebotar, M.; Martindale, W., III

doi:10.1090/S0002-9947-01-02731-3

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

Trans. Amer. Math. Soc. 353 (2001), 4235-4260

DOI: https://doi.org/10.1090/S0002-9947-01-02731-3

Published electronically: June 6, 2001

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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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