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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
DOI: https://doi.org/10.1090/S0002-9939-2012-11163-3
Published electronically: January 31, 2012
MathSciNet review: 2917078
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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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  • 1. P.M.Cohn, Universal Algebra, Harper & Row Publishers, New York-London, 1965. MR 0175948 (31:224)
  • 2. V.T.Filippov, On the theory of Mal$ '$cev algebras (Russian), Algebra i Logika, 16, no. 1 (1977), 101-108. MR 0506526 (58:22219)
  • 3. V.T.Filippov, Nilpotent ideals in Mal$ '$cev algebras (Russian), Algebra i Logika, 18, no. 5 (1979), 599-613. MR 582105 (82b:17024)
  • 4. N.Jacobson, Structure and Representations of Jordan Algebras, AMS Colloq. Publ. 39, AMS, Providence, RI, 1968. MR 0251099 (40:4330)
  • 5. E.N.Kuz$ '$min, Mal$ '$cev algebras and their representations (Russian), Algebra i Logika 7, no. 4 (1968), 48-69; English transl. in Algebra and Logic 7, no. 4 (1968), 233-244. MR 0252468 (40:5688)
  • 6. E.N.Kuz$ '$min and I.P.Shestakov, Nonassociative structures (Russian), Current problems in mathematics. Fundamental directions, Vol. 57, 179-266, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990; English transl. in Encyclopaedia of Math. Sciences 57, Algebra VI, eds. A.I. Kostrikin and I.R. Shafarevich. Springer-Verlag, 1995, 199-280. MR 1060322 (91i:17001)
  • 7. A.I.Mal$ '$cev, Analytic loops, Mat. Sb. (N.S.) 36 (78) (1955), 569-575. MR 0069190 (16:997g)
  • 8. S.V.Polikarpov and I.P.Shestakov, Nonassociative affine algebras (Russian), Algebra i Logika, 29, no. 6 (1990), 709-723; English transl. in Algebra and Logic 29, no. 6 (1990), 458-466. MR 1159142 (93b:17009)
  • 9. A.A.Sagle, Malcev algebras, Trans. Amer. Math. Soc. 101, no. 3 (1961), 426-458. MR 0143791 (26:1343)
  • 10. I.P.Shestakov, The radical and nilpotent elements of free alternative algebras (Russian), Algebra i Logika, 14, no. 3 (1975), 354-365; English transl. in Algebra and Logic, 14, no. 3 (1975), 219-226. MR 0427413 (55:447)
  • 11. I.P.Shestakov, On a problem by Shirshov (Russian), Algebra i Logika, 16, no. 2 (1977), 227-246; English transl. in Algebra and Logic, 16, no. 2 (1977), 153-166. MR 516039 (81c:17023)
  • 12. I.P.Shestakov, Finitely generated special Jordan and alternative PI-algebras (Russian), Mat. Sb. (N.S.) 122 (164) (1983), no. 1, 31-40. MR 715833 (84k:17018)
  • 13. I.P. Shestakov and E.I. Zelmanov, Prime alternative superalgebras and nilpotence of the radical of a free alternative algebra, Izv. Akad. Nauk SSSR Ser. Mat. 54, no. 4 (1990), 676-693. MR 1073082 (91j:17003)
  • 14. Ivan Shestakov and Natalia Zhukavets, The universal multiplicative envelope of the free Malcev superalgebra on one odd generator, Communications in Algebra 34, no. 4 (2006), 1319-1344. MR 2220815 (2007b:17051)
  • 15. E.I.Zel$ '$manov, Primary Jordan algebras (Russian), Algebra i Logika 18, no. 2 (1979), 162-175. MR 566779 (81k:17012)
  • 16. K.A.Zhevlakov, A.M.Slin$ '$ko, I.P.Shestakov, and A.I.Shirshov, Rings that are nearly associative (Russian), Moscow, Nauka, 1978; English translation by Academic Press, 1982, N.Y. MR 518614 (80h:17002)

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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: https://doi.org/10.1090/S0002-9939-2012-11163-3
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.

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