Skip to main content
Log in

An Elemental Characterization of Orthogonal Ideals in Lie Algebras

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we prove that the ideals generated by two elements x, y in a nondegenerate Lie algebra L over a ring of scalars Φ with \({\frac 1 2, \frac 1 3}\) are orthogonal if and only if [x, [y, L]] = 0.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Anquela J.A., Cortés T.: Local and subquotient inheritance of simplicity in Jordan systems. J. Algebra 240(2), 680–704 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anquela J.A., Cortés T., García E.: Local and subquotient inheritance of the heart of a Jordan system. Manuscripta Math. 106(3), 279–290 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anquela J.A., Cortés T., García E., McCrimmon K.: Outer inheritance of simplicity in Jordan systems. Comm. Algebra 32(2), 747–766 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anquela J.A., Cortés T., Loos O., McCrimmon K.: An elemental characterization of strong primeness in Jordan systems. J. Pure Appl. Algebra 109(1), 23–36 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beĭdar K.I., Mikhalëv A.V.: The structure of nondegenerate alternative algebras. Trudy Sem. Petrovsk. 243(12), 59–74 (1987)

    Google Scholar 

  6. Beĭdar K.I., Mikhalëv A.V., Slin’ko A.M.: A primality criterion for nondegenerate alternative and Jordan algebras. Trudy Moskov. Mat. Obshch. 261(50), 130–137 (1987)

    Google Scholar 

  7. Benkart G.: On inner ideals and ad-nilpotent elements of Lie algebras. Trans. Amer. Math. Soc. 232, 61–81 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brown B., McCoy N.H.: Prime ideals in nonassociative rings. Trans. Amer. Math. Soc. 89, 245–255 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  9. Calderón Martín, A.J., Forero Piulestán, M.: Inheritance of primeness by ideals in Lie triple systems. In: Algebras, Rings and Their Representations, pp. 7–16. World Science Publication, Hackensack (2006)

  10. Fernández Lopez A., García E., Gómez Lozano M.: The Jordan socle and finitary Lie algebras. J. Algebra 280(2), 635–654 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. García E.: Inheritance of primeness by ideals in Lie algebras. Int. J. Math. Game Theory Algebra 13(6), 481–484 (2003)

    MathSciNet  MATH  Google Scholar 

  12. García E., Gómez Lozano M.: An elemental characterization of strong primeness in Lie algebras. J. Algebra 312(1), 132–141 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. McCoy N.H.: Prime ideals in general rings. Am. J. Math. 71, 823–833 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  15. McCrimmon K.: Strong prime inheritance in Jordan systems. Algebras Groups Geom. 1(2), 217–234 (1984)

    MathSciNet  MATH  Google Scholar 

  16. Montaner F.: Local PI theory of Jordan systems. J. Algebra 216(1), 302–327 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Thedy A.: Z-closed ideals of quadratic Jordan algebras. Comm. Algebra 13(12), 2537–2565 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Esther García.

Additional information

J. Brox supported by the MEC through the FPU grant AP2009-4848, and partially supported by the Junta de Andalucía FQM264.

E. García partially supported by the MEC and Fondos FEDER MTM2010-16153, and by the Junta de Andalucía FQM 264.

M. Gómez Lozano partially supported by the MEC and Fondos FEDER MTM2010-19482, and by the Junta de Andalucía FQM264 and FQM3737.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brox, J., García, E. & Lozano, M.G. An Elemental Characterization of Orthogonal Ideals in Lie Algebras. Mediterr. J. Math. 11, 1061–1067 (2014). https://doi.org/10.1007/s00009-013-0344-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-013-0344-3

Mathematics Subject Classification (2010)

Keywords

Navigation