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Associo-symmetric algebras. (English) Zbl 0243.17002

Summary: 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 \(\neq 2\) is associative. In addition a finite-dimensional semisimple algebra in this class has an identity and is a direct sum of simple algebras.

MSC:

17A05 Power-associative rings
17A30 Nonassociative algebras satisfying other identities
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References:

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