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Prime Quotients of Jordan Systems and Lie Algebras

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Abstract

We show that, unlike alternative algebras, prime quotients of a nondegenerate Jordan system or a Lie algebra need not be nondegenerate, even if the original Jordan system is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts. Nevertheless, for Jordan systems and Lie algebras directly linked to associative systems, we prove that even semiprime quotients are necessarily nondegenerate.

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References

  1. Amitsur S.A.: A general theory of radicals. Am. J. Math. 76, 126–136 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  2. D’Amour A.: Quadratic Jordan systems of hermitian type. J. Algebra 149, 197–233 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anquela J.A., Cortés T.: Primitive Jordan pairs and triple systems. J. Algebra 184, 632–678 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anquela J.A., Cortés T.: Local and subquotient inheritance of simplicity in Jordan systems. J. Algebra 240, 680–704 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anquela, J.A., Cortés, T.: Growing hearts in associative systems. J. Algebra Appl. 11(5), 1250083 (11 p.) (2012)

  6. Anquela J.A., Cortés T.: Imbedding Lie algebras in strongly prime algebras. Commun. Algebra 41, 2467–2475 (2013)

    Article  MATH  Google Scholar 

  7. Anquela J.A., Cortés T., García E.: Herstein’s theorems and simplicity of hermitian Jordan systems. J. Algebra 246, 193–214 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Anquela J.A., Cortés T., Loos O., McCrimmon K.: An elemental characterization of strong primeness in Jordan systems. J. Pure Appl. Algebra 109, 23–36 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Anquela, J.A., Cortés, T.: McCrimmon, K.: Imbedding Jordan Systems in Primitive Systems (to appear)

  10. Baxter W.E., Martindale W.S. III: The extended centroid in *-prime rings. Commun. Algebra 10(8), 847–874 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Beidar K.I., Mikhalev A.V.: The structure of nondegenerate alternative algebras. J. Sov. Math. 47, 2525–2536 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. García E.: Tits–Kantor–Koecher algebras of strongly prime hermitian Jordan pairs. J. Algebra 277, 559–571 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. García E., Gomez Lozano M.: A characterization of the Kostrikin radical of a Lie algebra. J. Algebra 346, 266–283 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Herstein I.N.: Topics in Ring Theory. The University of Chicago Press, Chicago (1969)

    MATH  Google Scholar 

  15. Humphreys J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1980)

    MATH  Google Scholar 

  16. Jacobson, N.: Structure and Representations of Jordan Algebras, vol. 39. Am. Math. Soc. Colloq. Publ., Providence (1968)

  17. Jacobson N.: Lie Algebras. Dover Publications Inc., New York (1979)

    Google Scholar 

  18. Kostrikin A.I., Zelmanov E.I.: A theorem on sandwich algebras. Trudy Mat. Inst. Steklov 183, 106–111 (1990)

    MathSciNet  MATH  Google Scholar 

  19. Loos, O.: Jordan pairs. Lecture Notes in Mathematics, vol. 460. Springer, New York (1975)

  20. McCrimmon K.: On Herstein’s theorems relating Jordan and associative algebras. J. Algebra 13, 382–392 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  21. McCrimmon K.: Absolute zero divisors and local nilpotence in alternative algebras. Proc. Amer. Math. Soc 47(2), 293–299 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  22. McCrimmon K.: A Taste of Jordan Algebras. Springer, New York (2004)

    MATH  Google Scholar 

  23. McCrimmon K., Zelmanov E.: The structure of strongly prime quadratic Jordan algebras. Adv. Math. 69(2), 133–222 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Martindale W.S. III, Miers C.R.: Herstein’s Lie theory revisited. J. Algebra 98, 14–37 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Medvedev Yu.A.: Absolute zero divisors in special finitely generated Jordan algebras. Sib. Math. J. 29(2), 325–327 (1988)

    Article  MATH  Google Scholar 

  26. Medvedev, Yu.A., Zelmanov, E.I.: Some counter examples in the theory of Jordan algebras. In: González, S., Myung, H.C. (eds.) Nonassociative Algebraic Models (Universidad de Zaragoza, 1989), pp. 1–16. Nova Science Publishers, New York (1992)

  27. Pchelintsev S.V.: Prime algebras and absolute zero divisors. Math. USSR Izv. 28(1), 79–98 (1987)

    Article  MATH  Google Scholar 

  28. Shestakov I.P.: Absolute zero divisors and radicals of finitely generated alternative algebras. Algebra Log. 15(5), 582–602 (1976)

    Article  Google Scholar 

  29. Skosyrskii V.G.: Prime Jordan algebras and the Kantor construction. Algebra Log. 33(3), 169–179 (1995)

    Article  MathSciNet  Google Scholar 

  30. Strade H.: Simple Lie Algebras over Fields of Positive Characteristic. I Structure Theory. De Gruyter, Berlin (2004)

    Book  MATH  Google Scholar 

  31. Zelmanov E.I.: Absolute zero-divisors in Jordan pairs and Lie algebras. Math. USSR Sb. 40(4), 549–565 (1981)

    Article  Google Scholar 

  32. Zelmanov E.I.: Absolute zero-divisors and algebraic Jordan algebras. Sib. Math. J. 23(6), 841–854 (1982)

    Article  MathSciNet  Google Scholar 

  33. Zhevlakov K.A., Slinko A.M., Shestakov I.P., Shirshov A.I.: Rings that are Nearly Associative. Academic Press, New York (1982)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Oviedo, C/Calvo Sotelo s/n, 33007, Oviedo, Spain

    José A. Anquela & Teresa Cortés

  2. Departamento de Matemática Aplicada, Ciencia e Ingeniería de los Materiales y Tecnología Electrónica, Universidad Rey Juan Carlos, 28933, Móstoles, Madrid, Spain

    Esther García

  3. Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29071, Málaga, Spain

    Miguel Gómez Lozano

Authors
  1. José A. Anquela
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  2. Teresa Cortés
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  3. Esther García
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  4. Miguel Gómez Lozano
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Corresponding author

Correspondence to José A. Anquela.

Additional information

J. A. Anquela, T. Cortés, E. García: Partly supported by the Ministerio de Ciencia e Innovacion and Fondos FEDER, MTM2010-16153, the Ministerio de Economia y Competitividad and Fondos FEDER, MTM2014-52470-P, and the Junta de Andalucia FQM-264.

M. Gómez Lozano: Partly supported by the Ministerio de Ciencia e Innovacion and Fondos FEDER, MTM2010-19482, the Ministerio de Economia y Competitividad and Fondos FEDER, MTM2014-52470-P, and the Junta de Andalucia FQM-264.

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Anquela, J.A., Cortés, T., García, E. et al. Prime Quotients of Jordan Systems and Lie Algebras. Mediterr. J. Math. 13, 29–52 (2016). https://doi.org/10.1007/s00009-014-0488-9

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  • DOI: https://doi.org/10.1007/s00009-014-0488-9

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