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)
Previous Page Next Page