Alternator and associator ideal algebras

Abstract:

If I is the ideal generated by all associators, $(a,b,c) = (ab)c - a(bc)$, it is well known that in any nonassociative algebra $R,I \subseteq (R,R,R) + R(R,R,R)$. We examine nonassociative algebras where $I \subseteq (R,R,R)$. 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), (b, 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 $[x,(x,x,a)] = \gamma (x,x,[x,a])$ for fixed $\gamma$. 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.