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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
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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Keywords: Associo-symmetric, power-associative, orthogonal idempotents, semisimple, degree, principal idempotent
Article copyright: © Copyright 1972 American Mathematical Society