Alternator and Associator Ideal Algebras

Abstract

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.