Right alternative rings of characteristic two

Abstract

for all w, x, y and showed by example that (1.1) can fail to hold. Prior to this, Kleinfeld [1 ] generalized the Skornyakov theorem in another direction by assuming only the absence of one sort of nilpotent element. We now specify Kleinfeld's result in detail. Let F be the free nonassociative ring generated by xi and x2 and suppose that R is any right alternative ring. Kleinfeld calls t, u, v in R an alternative triple if (i) there exist elements a [xl, X2], 3 [Xl, x2], 7Y [xl, X2] in F and elements ri, r2 in R such that t =a [ri, r2 ], u =-1 [ri, r2], v = y [ri, r2] and (ii) if si and S2 are elements from an arbitrary alternative ring, and if t'=a [sl, S2], u'=3[sl, S2], v'-=Y[sl, S2], then (t', u', v') =0. The ring R is said to have property (P) if t, u, v an alternative triple in R and (t, u, V)2=0 imply (t, u, v) =0. By the definition of an alternative triple, an alternative ring has property (P). Kleinfeld's result is the converse, assuming characteristic not two; that is, a right alternative ring of characteristic not two is alternative if (and only if) it has property (P). We herein extend this line of investigation by proving that a right alternative ring of characteristic two, satisfying (1.1), is alternative if (and only if) it has property (P). The methods are mainly those used in [2], coupled with two essential lemmas (numbered 4 and 5 in our paper) due to Kleinfeld. Following [2], we say that R is strongly right alternative if R is a right alternative ring satisfying (1.1) . Throughout the paper, R will always denote such a ring, with the additional hypothesis that R have characteristic two.