M. L. RACINE

MARTINDALE'S THEOREM ([26]). Let (0, *) be an associativ e algebra

n

with involution. Assume that 1 = / e., n 3, where the e. are orthogonal

• - i

l l

1 - 1 r k k k

idempotents such that e,

€

M(G, *), G.e.ci= G and e. =J a,, b a., for some

k k k k

i ^ i , a e e ne.3 a = (a .)* and b.

e

M(G. *) . Then any homomorphism of

ij i ] ji i] J

W(G, *) into an algebra B , where B is a (unital) associativ e algebra, ha s a

unique extension of G into B.

If (G, *) is central simple over a field $ and the degree of G 3,

extending the bas e field if necessary , we may apply Martindale's Theorem.

In particular we obtain that M(G, *) generates G; an automorphism of G is

either an automorphism of G or an antiautomorphism of G. If G/$ is central

simple, the automorphisms of M(G, *) are inner automorphisms of

G : a - c a c with c c * € $ 1 . If G/r is central simple, r / § a quadratic

field extension, the automorphisms of W(G, *) are automorphisms of G/$,

which commute with *; thes e need not be inner. (See [16], p . 248, where

the arguments are als o valid in characteristic 2. )

§2. Orders in Jordan Algebras.

Let K be the quotient field of a Dedekind domain o. If v is a finite

dimensional K-vector space , an o-lattice L of V is a finitely generated

o-module L C As with KL = V. An element x of a Jordan algebra over K is

said to be integral if it satisfie s a monic integral polynomial, that is with

coefficients in o. From now on in this chapter $ will be a finite dimension -

al Jordan algebra over K. We note that (4) and (5) imply that o[x], a s an