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The Prime Radical of Alternative Rings and Loops

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Abstract

A characterization of the prime radical of loops as the set of strongly Engel elements was given in our earlier paper. In this paper, some properties of the prime radical of loops are considered. Also a connection between the prime radical of the loop of units of an alternative ring and the prime radical of this ring is given.

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References

  1. K. I. Beidar, A. V. Mikhalev, and A. M. Slin’ko, “A primality criterion for nondegenerate alternative and Jordan algebras,” Tr. Mosk. Mat. Obshch., 50, 130–137 (1987).

    MATH  Google Scholar 

  2. V. D. Belousov, Foundation of the Theory of Quasigroups and Loops [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  3. R. Bruck, A Survey of Binary Systems, Springer, Berlin (1958).

    Book  MATH  Google Scholar 

  4. E. G. Goodaire, E. Jespers, and C. Polcino Miles, Alternative Loop Rings, Elsevier, Amsterdam (1996).

    MATH  Google Scholar 

  5. A. V. Gribov and A. V. Mikhalev, “Prime radical of loops and Ω-loops. I,” Fundam. Prikl. Mat., 19, No. 2, 25–42 (2014).

    MathSciNet  Google Scholar 

  6. A. G. Kurosh, “Radicals of rings and algebras,” Mat. Sb., 33, No. 1, 13–26 (1953).

    MathSciNet  MATH  Google Scholar 

  7. J. Levitzki, “Prime ideals and the lower radical,” Amer. J. Math., 73, 25–29 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Pflugfelder, Quasigroups and Loops: Introduction, Sigma Ser. Pure Math., Vol. 7, Heldermann (1991).

  9. M. Rich, “Some radical properties of s-rings,” Proc. Am. Math. Soc., 30, No. 1, 40–42 (1971).

    MathSciNet  MATH  Google Scholar 

  10. M. Rich, “The prime radical in alternative rings,” Proc. Am. Math. Soc., 56, 11–15 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  11. K. K. Shukin, “RI*-solvable radical of groups,” Mat. Sb., 52, No. 4, 1021–1031 (1960).

    MathSciNet  Google Scholar 

  12. L. A. Skornyakov, “Right-alternative fields,” Izv. Akad. Nauk SSSR, Ser. Mat., 15, No. 2, 177–184 (1951).

    MathSciNet  MATH  Google Scholar 

  13. D. Stanovsky and P. Vojtechovsky, “Commutator theory for loops,” J. Algebra, 399, 290–322 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Tsai, “The prime radical in a Jordan ring,” Proc. Am. Math. Soc., 19, 1171–1175 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  15. E. I. Zelmanov, “Prime alternative superalgebras and the nilpotence of the radical of a free alternative algebra,” Izv. Akad. Nauk SSSR, Ser. Mat., 54, No. 4, 676–693 (1990).

    Google Scholar 

  16. K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings Close to Associative [in Russian], Nauka, Moscow (1978).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    A. V. Gribov

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  1. A. V. Gribov
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Correspondence to A. V. Gribov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 1, pp. 145–166, 2015.

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Gribov, A.V. The Prime Radical of Alternative Rings and Loops. J Math Sci 223, 587–601 (2017). https://doi.org/10.1007/s10958-017-3368-6

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  • DOI: https://doi.org/10.1007/s10958-017-3368-6

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