Alternator and associator ideal algebras

Abstract

Abstract. If / is the ideal generated by all associators, (a, b, c) = (ab)ca{bc), it is well known that in any nonassociative algebra R, I C (R, R, R) + R(R, R, R). We examine nonassociative algebras where / C (R, R, R). Such algebras include (-1, 1) algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), {a, b, a), (f>, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where I = {R, R, R) as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form [x, (x, x, a)] = y(x, x, [x, a]) for fixed y. With two exceptions, if this algebra has an idempotent e such that (e, e, R) = 0, then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.

Alternators of a Right Alternative Algebra

“…In any nonassociative algebra, the associator (a, b, c) and the commutator [a, b] We show exactly what identities are required to make M a left ideal, and we give an easy way to deduce if a given right alternative algebra satisfies them. Some work [3] has been done on the structure of right alternative algebras where M is assumed to be an ideal. Till now, the actual strength of this assumption was not known.…”