Alternative rings without nilpotent elements
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- by Irvin Roy Hentzel
- Proc. Amer. Math. Soc. 42 (1974), 373-376
- DOI: https://doi.org/10.1090/S0002-9939-1974-0327858-3
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Abstract:
In this paper we show that any alternative ring without nonzero nilpotent elements is a subdirect sum of alternative rings without zero divisors. Andrunakievic and Rjabuhin proved the corresponding result for associative rings by a complicated$^{1}$ process in 1968. Our result extends Andrunakievic and Rjabuhin’s result to the alternative case, and our argument is nearly as simple as in the associative-commutative case. Since right alternative rings of characteristic not 2 without nilpotent elements are alternative, our results extend to such rings as well.References
- V. A. Andrunakievič and Ju. M. Rjabuhin, Rings without nilpotent elements, and completely prime ideals, Dokl. Akad. Nauk SSSR 180 (1968), 9–11 (Russian). MR 0230760
- Ernst-August Behrens, Ring theory, Pure and Applied Mathematics, Vol. 44, Academic Press, New York-London, 1972. Translated from the German by Clive Reis. MR 0379551
- R. H. Bruck and Erwin Kleinfeld, The structure of alternative division rings, Proc. Amer. Math. Soc. 2 (1951), 878–890. MR 45099, DOI 10.1090/S0002-9939-1951-0045099-9
- Erwin Kleinfeld, Right alternative rings, Proc. Amer. Math. Soc. 4 (1953), 939–944. MR 59888, DOI 10.1090/S0002-9939-1953-0059888-X
- Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR 0210757
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 373-376
- MSC: Primary 17D05
- DOI: https://doi.org/10.1090/S0002-9939-1974-0327858-3
- MathSciNet review: 0327858