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Transactions of the American Mathematical Society
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
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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0310025-X
PII: S 0002-9947(1972)0310025-X
Keywords: Associo-symmetric, power-associative, orthogonal idempotents, semisimple, degree, principal idempotent
Article copyright: © Copyright 1972 American Mathematical Society