A remark on a theorem of Marshall Hall

Abstract

characterizes quaternion algebras among associative but not commutative division rings. Our remark is that (1) characterizes CayleyDickson algebras among alternative but not associative division rings. This follows from Hall's proof and a result of A. A. Albert [1]. A theorem of R. D. Schafer [4, Theorem 4] permits us to conclude that (1) and Hall's Theorem L [2] (as universal) ensure that the coordinate ring of a projective plane is uniquely defined irrespective of the coordinate system. It is easy to verify (1) in a Cayley-Dickson algebra. We should like to sketch a proof of the converse, which is independent of Albert's result in [1]. Let A be an alternative division ring and let F be the set of all elements c CA which satisfy cx = xc for every x EA. Then F is a field2 and, when 3 is nonzero in A, F is the center of A [3; 5, Lemma 9]. If A satisfies (1), the proof of Hall's Lemma 1 [2, p. 262] yields x2=t(x)x-n(x) for every xzA not in F, where t(x), n(x)EF. Define t(c) = 2c, n(c) = c2 for cC F. When 3 is zero in A and A satisfies (1), F is still the center of A as we shall now show.3 Since 2 is a nonzero in A, for each xCA we have (x+c)2CF for some cCF. If, then, cl, c2 are in F, we have (cl, c2, (x+c)2)=0. By use of some identities of Zorn [6, (1.6) and (1.8); 5, (1)] we find that (Cl, C2, (X+C)2)=2(x+c)(c1, C2, x+c)=2(x+c) * (cl, c2, x). We infer that (cl, c2, x) = 0 for every cl, c2 in F and every x in A. Now let cEF and x, yEA. As before, (x+c')2CF for some c'EF and