A generalization of alternative rings

Abstract

From our earlier remarks it is immediate that these rings are generalizations of alternative rings. In ?2 we show that any ring A satisfying (1), (2), and (3) must be powerassociative and, using this result we obtain an idempotent decomposition for A as A = A1 + A112 + Ao where x E Ai if and only if ex + xe = 2ix for the idempotent e of A. In Theorem 3 we develop some fundamental relations for the multiplicative properties of the Ai. We are able to show in ?3 that if A has no nil ideals then A must, in fact, have a Peirce decomposition with respect to an idempotent e. That is, A is the direct sum of the subgroups Aij; i, j = 0, 1 where x E A if and only if ex = ix, xe = jx. This is then used to prove the main results: (a) Any simple ring A satisfying (1), (2), and (3) with an idempotent e : 1 must be associative or a Cayley-Dickson algebra over its center. (b) Any finite-dimensional semi-simple algebra A satisfying (1), (2), and (3) has a unity element and is the direct sum of simple algebras. In ?5 we give some examples to show that these results are in a certain sense best possible. We suppose in the remainder of this paper that the ring A satisfies (1), (2), and(3).