Generalization of right alternative rings

Abstract

Let R be a ring of characteristic not two, satisfying the following three identities, which are consequences of the right alternative identity: ( x, x, x) = 0, ( wx, y, z) + ( w, x, ( y, z)) = w( x, y, z) + ( w, y, z) x, and ( y, y, x) k = 0, for some positive integer k = k( x, y). A simple ring in this class which is not associative is alternative if and only if R has an idempotent e such that there are no nilpotent elements in R 1( e) and R 0( e), the zero and one subspaces of the Albert decomposition. This generalizes the comparable result for right alternative rings by Humm-Kleinfeld.