Rings Satisfying (x, y, z) = (y, z, x)

Abstract

Let R be a ring satisfying the identity1for all x, y, z ϵ R, where (x,y,z) = (xy)z — x(yz). If R also satisfies the identity (x, x, x) = 0 for all x ϵ R, then R is alternative. It is known that if R satisfies (1), it need not be an alternative (see 6). Thus, the class of rings satisfying (1) is a non-trivial extension of the class of alternative rings. P. Jordan remarked that (x, x, x)2 = 0 is an identity in R (see 9).