(-1,1) algebras

Abstract

In a nonassociative ring A, the symbol (a, 6, c) where a, b, c are elements of A is defined as (a, b, c) = (ab)c -a(bc). The symbol [a, b] where a, 6 are elements of A is defined as [a, b] =ab -ba. The nucleus of A, N(A)= {nEA\ (re, a, 6) = (a, n, 6) = (a, 6, n) = 0 for all a, bEA ). The center of A, C(A) = {sEN(A)\ [s, a] =0 for all aEA). A trivial ideal of A is an ideal ly±0 of A such that P = 0.A ( -1, 1) ring A is a nonassociative ring in which the following identities are assumed to hold.(1) (a, b, c) + (a, c, b) = 0,for all a, 6, c elements of A. A ( -1, 1) algebra is a ( -1, 1) ring with identity which is also a finite dimensional vector space over a field F which satisfies a(ab) =a(ab) = (aa)b for all a, 6 elements of A, a in F. We shall not require that a subalgebra of A contain the identity of A, though we do require that it contain an identity of its own. When hypotheses are placed on a ring as a whole, often these hypotheses imply certain properties for the center. For example, the center of a simple ring is a field. It is possible then, that hypotheses placed on the center will be reflected in the structure of the whole ring. What hypotheses are possible? Keeping in mind that the center of a simple ring is a field, one might suggest we assume that the center is simple, or semisimple. We could assume there are no ideals of the whole ring contained in the center. Or, we might assume that the center has no nilpotent elements. In this paper we show that any one of these conditions is sufficient to make a ( -1, 1) algebra associative. We prove the theorem:Theorem.If A is a ( -1, 1) algebra over afield of characteristic 9^2, 3, and the center of A contains no trivial A ideals, then A is associative.In this proof we use Albert's result that semisimple finite dimensional right alternative algebras are alternative [2], and Wedderburn's structure theorem that semisimple associative algebras are complete direct sums of matrix rings over division rings [3]. We use Maneri's result that for a ( -1, 1) algebra of characteristic 9^2, 3, (A, A, A) is an ideal [5], and we use the results in my paper [4] in several places.