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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the radical of a free Malcev algebra


Authors: I. P. Shestakov and A. I. Kornev
Journal: Proc. Amer. Math. Soc. 140 (2012), 3049-3054
MSC (2010): Primary 17D10, 17D05, 17A50, 17A65
Published electronically: January 31, 2012
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Abstract: We prove that the prime radical $ rad\,\mathcal {M}$ of the free Malcev algebra $ \mathcal {M}$ of rank more than two over a field of characteristic $ \neq 2$ coincides with the set of all universally Engelian elements of $ \mathcal {M}$. Moreover, let $ T(\mathbb{M})$ be the ideal of $ \mathcal {M}$ consisting of all stable identities of the split simple 7-dimensional Malcev algebra $ \mathbb{M}$ over $ F$. It is proved that $ rad\,\mathcal {M}=J(\mathcal {M})\cap T(\mathbb{M})$, where $ J(\mathcal {M})$ is the Jacobian ideal of $ \mathcal {M}$. Similar results were proved by I. Shestakov and E. Zelmanov for free alternative and free Jordan algebras.


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Additional Information

I. P. Shestakov
Affiliation: Institute of Mathematics and Statistics, University of São Paulo, Rua do Matao, 1010, Cidade Universitária, São Paulo 05508-090, Brazil

A. I. Kornev
Affiliation: IMECC Cidade Universitária Zeferino Vaz, Campinas, 13083-859 São Paulo, Brazil
Address at time of publication: Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Rua Santa Adélia, 166, Blocoa, Bairro Bangu, Santo André, SP, Brazil 09210-170

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11163-3
PII: S 0002-9939(2012)11163-3
Keywords: Malcev algebra, free algebra, prime radical, nilpotent element, Engelian element.
Received by editor(s): February 23, 2011
Received by editor(s) in revised form: March 31, 2011
Published electronically: January 31, 2012
Additional Notes: The first author was supported by FAPESP grant 2010/50347-9 and CNPq grant 305344/ 2009-9
The second author was supported by FAPESP grant 2008/57680-5
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.