zo M. L. RACINE

II. MAXIMAL ORDERS IN SPECIAL JORDAN ALGEBRAS

§1. Orders in Special Jordan Algebras.

Let K be the quotient field of a Dedekind ring o, ^ a finite dimen-

sional special Jordan algebra over K. The following lemma will be fundamen-

tal in what follows; it permits passag e from Jordan orders to associativ e

orders.

LEMMA 1. Let ^ be an arbitrary special Jordan algebra over K and

assume J C G , where Q is associative . If M is an order of $ then M'

the (associative) o-algebra generated by M in G is an (associative) order

of $x the associativ e K-algebra generated by ^ i-n G.

PROOF. Let x , , . . . , x be a set of generators of M. Then M' is

1 m

spanned by elements of the form x, x. . . . x. , x. e {x,, . . . , x }. We

in i

0

i

I.

1 m

1 2 r }

claim that M' is spanned by elements of the form x. x. . . . x. with

1 2 r

i i_ . . . i (therefore r m and this is a finite set) . Induct on the

1 2 r —

2

length of a monomial. Since M is closed under squaring (x = 1U ),

2 2 2

x, y € M implies xy + yx = (x + y) - x - y

€

M. If x. . . . x is an

2 l r

arbitrary monomial and x. = x. , x. e M and the monomial can be written

J J+1 J

a s a sum of integers times monomials of shorter length. If i. i.

+ 1

,

x, x, = -x. x. + terms in M. Repeating the process one obtains

i. i. ,,

I.

,-,

i.

monomials of shorter length and monomials of the desired form. Therefore

M' is a finitely generated o-module containing 1. It is clearly a ring and

since KM = £ K(M') = (KM)' = ?. Therefore M' is an (associative)