Alternators of a right alternative algebra

Abstract:

We show that in any right alternative algebra, the additive span of the alternators is nearly an ideal. We give an easy test to use to determine if a given set of additional identities will imply that the span of the alternators is an ideal. We apply our technique to the class of right alternative algebras satisfying the condition $(a,a,b) = \lambda [a,[a,b]]$. We show that any semiprime algebra over a field of characteristic $\ne 2$, $\ne 3$ which satisfies the right alternative law and the above identity with $\lambda \ne 0$ is a subdirect sum of (associative and commutative) integral domains.