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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Associo-symmetric algebras


Authors: Raymond Coughlin and Michael Rich
Journal: Trans. Amer. Math. Soc. 164 (1972), 443-451
MSC: Primary 17A30
DOI: https://doi.org/10.1090/S0002-9947-1972-0310025-X
MathSciNet review: 0310025
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Abstract: Let A be an algebra over a field F satisfying $ (x,x,x) = 0$ with a function $ g:A \times A \times A \to F$ such that $ (xy)z = g(x,y,z)x(yz)$ for all x, y, z in A. If $ g({x_1},{x_2},{x_3}) = g({x_{1\pi }},{x_{2\pi }},{x_{3\pi }})$ for all $ \pi $ in $ {S_3}$ and all $ {x_1},{x_2},{x_3}$ in A then A is called an associo-symmetric algebra. It is shown that a simple associo-symmetric algebra of degree $ > 2$ or degree $ = 1$ over a field of characteristic $ \ne 2$ is associative. In addition a finite-dimensional semisimple algebra in this class has an identity and is a direct sum of simple algebras.


References [Enhancements On Off] (What's this?)

  • [1] A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc. 64 (1948), 552-593. MR 10, 349. MR 0027750 (10:349g)
  • [2] -, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc. 69 (1950), 503-527. MR 12, 475. MR 0038959 (12:475d)
  • [3] L. A. Kokoris, New results on power-associative algebras, Trans. Amer. Math. Soc. 77 (1954), 363-373. MR 16, 442. MR 0065543 (16:442b)
  • [4] F. Kosier, On a class of nonflexible algebras, Trans. Amer. Math. Soc. 102 (1962), 299-318. MR 24 #A3187. MR 0133353 (24:A3187)
  • [5] R. H. Oehmke, Commutative power-associative algebras of degree one, J. Algebra 14 (1970), 326-332. MR 0286851 (44:4058)
  • [6] R. D. Schafer, An introduction to nonassociative algebras, Pure and Appl. Math., vol. 22, Academic Press, New York, 1966. MR 35 #1643. MR 0210757 (35:1643)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0310025-X
Keywords: Associo-symmetric, power-associative, orthogonal idempotents, semisimple, degree, principal idempotent
Article copyright: © Copyright 1972 American Mathematical Society

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