Scalar dependent algebras in the alternative sense

Joyce Longman and Michael Rich

Vol. 76 (1978), No. 2, 463–470

DOI: 10.2140/pjm.1978.76.463

Abstract

Let R, a not necessarily associative algebra over a field F of characteristic ≠2, be equipped with a map g : R × R × 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.

Mathematical Subject Classification 2000

Primary: 17D05

Secondary: 17A30

Milestones

Received: 3 May 1977

Revised: 2 November 1977

Published: 1 June 1978