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 ) = λ [ a , [ a , b ] ] (a,a,b) = \lambda [a,[a,b]] . We show that any semiprime algebra over a field of characteristic ≠ 2 \ne 2 , ≠ 3 \ne 3 which satisfies the right alternative law and the above identity with λ ≠ 0 \lambda \ne 0 is a subdirect sum of (associative and commutative) integral domains.