Published online by Cambridge University Press: 20 November 2018
Bruck and Kleinfeld [3] proved that any alternative ring with characteristic prime to 2 must satisfy the identity
where the associator (x,y,z) is defined by (x, y, z) = (xy)z – x(yz) and . Linearization of the identity (x2, y, z) = 2x · (x, y, z) yields for characteristic prime to 2 an equivalent identity
(1)
Using the right alternative law (x, y, z) = –(y, x, z) and the flexible law (x, y, z) = –(z, y, x) which is satisfied in any alternative ring we obtain
(2)
and
(3)