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Finite power-associative division rings. (English) Zbl 0145.25701

The author proves the well-known theorem: “A finite strictly power-associative division ring of characteristic \(\neq 2\) is a field” without using the classification of central simple Jordan algebras, as in Albert’s proof [cf. R. D. Schafer, An introduction to nonassociative algebras. New York etc.: Academic Press (1966; Zbl 0145.25601), p. 133–136]. He applies the methods of Shirshov and Cohn, dealing with a special Jordan algebra of symmetric elements, induced by an associative algebra with involution.

MSC:

17A05 Power-associative rings
17A35 Nonassociative division algebras

Citations:

Zbl 0145.25601
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Full Text: DOI

References:

[1] A. A. Albert, On nonassociative division algebras, Trans. Amer. Math. Soc. 72 (1952), 296 – 309. · Zbl 0046.03601
[2] A. A. Albert, A construction of exceptional Jordan division algebras, Ann. of Math. (2) 67 (1958), 1 – 28. · Zbl 0079.04701 · doi:10.2307/1969922
[3] A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc. 64 (1948), 552 – 593. · Zbl 0033.15402
[4] A. A. Albert, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc. 69 (1950), 503 – 527. · Zbl 0039.26501
[5] E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. · Zbl 0077.02101
[6] N. Jacobson, Associative algebras with involution and Jordan algebras, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 202 – 212. · Zbl 0136.02102
[7] N. Jacobson, A coordinatization theorem for Jordan algebras, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1154 – 1160. · Zbl 0115.02703
[8] Louis A. Kokoris, New results on power-associative algebras, Trans. Amer. Math. Soc. 77 (1954), 363 – 373. · Zbl 0057.27203
[9] Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. · Zbl 0145.25601
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