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Prime Algebras Connected With Monsters

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Abstract

We study the overalgebras and the ideals of the Jordan algebras possessing prime (−1, 1)-envelopings. If a Jordan algebra possesses a prime nonassociative (−1, 1)-enveloping then we prove that it is also prime; furthermore, its every ideal is a prime algebra. In particular, the overalgebras and metaideals of Jordan monsters are prime.

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References

  1. Pchelintsev S. V., “Nilpotent elements and nil-radicals of alternative algebras,” Algebra and Logic, vol. 24, no. 6, 441–454 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  2. Pchelintsev S. V., “Prime algebras and absolute zero divisors,” Math. USSR Izv., vol. 28, no. 1, 79–98 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  3. McCrimmon K., A Taste of Jordan Algebras, Springer-Verlag, New York (2004).

    MATH  Google Scholar 

  4. Medvedev Yu. A. and Zelmanov E. I., “Some counterexamples in the theory of Jordan algebras,” in: Nonassociative Algebraic Models (Zaragoza, 1989), Nova Sci. Publ., Commack, New York, 1992, 1–16.

    Google Scholar 

  5. Shestakov I. P., “Superalgebras and counterexamples,” Sib. Math. J., vol. 32, no. 6, 1052–1060 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  6. Pchelintsev S. V. and Shestakov I. P., “Prime (-1, 1) and Jordan monsters and superalgebras vector type,” J. Algebra, vol. 423, 54–86 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhevlakov K. A., Slin'ko A. M., Shestakov I. P., and Shirshov A. I., Rings That Are Nearly Associative, Academic Press, New York (1982).

    MATH  Google Scholar 

  8. Hentzel I. R., “The characterization of (-1, 1)-rings,” J. Algebra, vol. 30, 236–258 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  9. Filippov V. T., “Prime Mal'tsev algebras,” Math. Notes, vol. 31, no. 5, 341–345 (1982).

    Article  MATH  Google Scholar 

  10. Zelmanov E. I., “On prime Jordan algebras. II,” Sib. Math. J., vol. 24, no. 1, 89–104 (1983).

    MathSciNet  Google Scholar 

  11. Skosyrsky V. G., “Right alternative algebras,” Algebra and Logic, vol. 23, no. 2, 131–136 (1984).

    Article  MathSciNet  Google Scholar 

  12. Pchelintsev S. V., “Prime alternative algebras that are nearly commutative,” Izv. Math., vol. 68, no. 1, 181–204 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  13. Pchelintsev S. V., “Exceptional prime alternative algebras,” Sib. Math. J., vol. 48, no. 6, 1060–1073 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  14. Pchelintsev S. V., “Degenerate alternative algebras,” Sib. Math. J., vol. 55, no. 2, 323–335 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  15. Pchelintsev S. V., “Isotopes of the alternative monster and the Skosyrsky algebra,” Sib. Math. J., vol. 57, no. 4, 666–678 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  16. Pchelintsev S. V., “Isotopes of prime (-1, 1)-and Jordan algebras,” Algebra and Logic, vol. 49, no. 3, 262–288 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  17. Pchelintsev S. V., “On the commutator nilpotency step of strictly (-1, 1)-algebras,” Math. Notes, vol. 93, no. 5, 756–762 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  18. Pchelintsev S. V., “Commutator identities of the homotopes of (-1, 1)-algebras,” Sib. Math. J., vol. 54, no. 2, 325–340 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  19. Kleinfeld E., “Right alternative rings,” Proc. Amer. Math. Soc., vol. 4, 939–944 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  20. Albert A. A., “Almost alternative algebras,” Port. Math., vol. 8, 23–36 (1949).

    MathSciNet  MATH  Google Scholar 

  21. Kleinfeld E., “On a class of right alternative rings,” Math. Z., Bd 87, 12–16 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  22. Roomel'di R. E., “Centers of a free (-1, 1)-ring,” Sib. Math. J., vol. 18, no. 4, 610–622 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  23. Hentzel I. R., “Nil semi-simple (-1, 1)-rings,” J. Algebra, vol. 22, no. 3, 442–450 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  24. Shestakov I. P., “Prime alternative superalgebras of an arbitrary characteristic,” Algebra and Logic, vol. 36, no. 6, 701–731 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  25. McCrimmon K., “Speciality and non-speciality of two Jordan superalgebras,” J. Algebra, vol. 149, no. 2, 326–351 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  26. Skosyrsky V. G., “Prime Jordan algebras and the Kantor construction,” Algebra and Logic, vol. 33, no. 3, 169–179 (1994).

    Article  MathSciNet  Google Scholar 

  27. Hentzel I. R., “Nil semi-simple locally (-1, 1)-rings,” Bull. Iran. Math. Soc., vol. 9, 11–14 (1981).

    MathSciNet  Google Scholar 

  28. Pchelintsev S. V., “Manifolds generated by free algebras of type (-1, 1) of finite rank,” Sib. Math. J., vol. 28, no. 2, 299–306 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  29. Skosyrsky V. G., “Strongly prime noncommutative Jordan algebras,” in: Studies on the Theory of Rings and Algebras [Russian], Nauka, Novosibirsk, 1989, vol. 16, 131–164.

    MathSciNet  Google Scholar 

  30. Zelmanov E. I., “Jordan algebras with finiteness conditions,” Algebra and Logic, vol. 17, no. 6, 450–457 (1979).

    Article  MathSciNet  Google Scholar 

  31. Pchelintsev S. V., “On central ideals of finitely generated binary (-1, 1)-algebras,” Sb. Math., vol. 193, no. 4, 585–607 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  32. Roomel'di R. E., “The lower nil radical of (-1, 1) rings,” Algebra and Logic, vol. 12, no. 3, 184–189 (1973).

    Article  MATH  Google Scholar 

  33. Pchelintsev S. V., Structure and Identities of Nilalgebras That Are Close to Associative [Russian], Diss. Dokt. Fiz.-Mat. Nauk, Moscow (1986).

    Google Scholar 

  34. Pchelintsev S. V., “Identities of a free (-1, 1)-algebra of rank 3,” in: Studies on the Theory of Rings and Algebras. Vol. 16 [Russian], Nauka, Novosibirsk, 1989, 110–131.

    Google Scholar 

  35. Maneri C., “Simple (-1, 1)-rings with an idempotent,” Proc. Amer. Math. Soc., vol. 14, 110–117 (1963).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Financial University Under the Government of the Russian Federation, Moscow, Russia

    S. V. Pchelintsev

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    S. V. Pchelintsev

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  1. S. V. Pchelintsev
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Correspondence to S. V. Pchelintsev.

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Original Russian Text Copyright © 2018 Pchelintsev S.V.

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Pchelintsev, S.V. Prime Algebras Connected With Monsters. Sib Math J 59, 341–356 (2018). https://doi.org/10.1134/S0037446618020179

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  • DOI: https://doi.org/10.1134/S0037446618020179

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