Scalar dependent algebras in the alternative sense

Let R, a not necessarily associative algebra over a field F of characteristic Φ 2, be equipped with a map g\ R X R X R->F. We show that if R contains a nonzero idempotent and satisfies the identities (1) (xy)z + (yx)z — g(x, y, z)[x(yz) + y(xz)] and (2) (xy)z + (xz)y = g(x, y, z)[x{yz) + x(zy)] then...

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Journal Title: Pacific Journal Of Mathematics Vol. 76; no. 2; pp. 463 - 470
Authors: Joyce Longman, Michael Rich
Format: Article
Published: 1978
Subjects:
Online Access: Full Text
Summary: Let R, a not necessarily associative algebra over a field F of characteristic Φ 2, be equipped with a map g\ R X R X R->F. We show that if R contains a nonzero idempotent and satisfies the identities (1) (xy)z + (yx)z — g(x, y, z)[x(yz) + y(xz)] and (2) (xy)z + (xz)y = g(x, y, z)[x{yz) + x(zy)] then R is an alternative algebra. The methods also apply to other pairs of identities.
ISSN: 00308730