On the radical of a free Malcev algebra
HTML articles powered by AMS MathViewer
- by I. P. Shestakov and A. I. Kornev PDF
- Proc. Amer. Math. Soc. 140 (2012), 3049-3054 Request permission
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.References
- P. M. Cohn, Universal algebra, Harper & Row, Publishers, New York-London, 1965. MR 0175948
- V. T. Filippov, On the theory of Mal′cev algebras, Algebra i Logika 16 (1977), no. 1, 101–108, 125 (Russian). MR 0506526
- V. T. Filippov, Nilpotent ideals in Mal′cev algebras, Algebra i Logika 18 (1979), no. 5, 599–613, 633 (Russian). MR 582105
- Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099
- E. N. Kuz′min, Mal′cev algebras and their representations, Algebra i Logika 7 (1968), no. 4, 48–69 (Russian). MR 0252468
- E. N. Kuz′min and I. P. Shestakov, Nonassociative structures, Current problems in mathematics. Fundamental directions, Vol. 57 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp. 179–266 (Russian). MR 1060322
- A. I. Mal′cev, Analytic loops, Mat. Sb. N.S. 36(78) (1955), 569–576 (Russian). MR 0069190
- S. V. Polikarpov and I. P. Shestakov, Nonassociative affine algebras, Algebra and Logic 29 (1990), no. 6, 458–466 (1991). MR 1159142, DOI 10.1007/BF01978558
- Arthur A. Sagle, Malcev algebras, Trans. Amer. Math. Soc. 101 (1961), 426–458. MR 143791, DOI 10.1090/S0002-9947-1961-0143791-X
- I. P. Šestakov, Radicals and nilpotent elements of free alternative algebras, Algebra i Logika 14 (1975), no. 3, 354–365, 370 (Russian). MR 0427413
- I. P. Šestakov, A problem of Širšov, Algebra i Logika 16 (1977), no. 2, 227–246, 251 (Russian). MR 516039
- I. P. Shestakov, Finitely generated special Jordan and alternative PI-algebras, Mat. Sb. (N.S.) 122(164) (1983), no. 1, 31–40 (Russian). MR 715833
- E. I. Zel′manov and I. P. Shestakov, Prime alternative superalgebras and the nilpotency of the radical of a free alternative algebra, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 676–693 (Russian); English transl., Math. USSR-Izv. 37 (1991), no. 1, 19–36. MR 1073082
- Ivan Shestakov and Natalia Zhukavets, The universal multiplicative envelope of the free Malcev superalgebra on one odd generator, Comm. Algebra 34 (2006), no. 4, 1319–1344. MR 2220815, DOI 10.1080/00927870500454570
- E. I. Zel′manov, Primary Jordan algebras, Algebra i Logika 18 (1979), no. 2, 162–175, 253 (Russian). MR 566779
- K. A. Ževlakov, A. M. Slin′ko, I. P. Šestakov, and A. I. Širšov, Kol′tsa, blizkie k assotsiativnym, Sovremennaya Algebra. [Modern Algebra], “Nauka”, Moscow, 1978 (Russian). MR 518614
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
- MR Author ID: 289548
- 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
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2917078