ARITHMETICS OF JORDAN ALGEBRAS 9

o-module, is spanned by the powers of x. The following proposition is an

immediate consequence of the above definitions.

PROPOSITION 1. The following are equivalent.

(i) x € P is integral.

(ii) o[x] is a finitely generated o-module.

(iii) The minimum polynomial of x is integral.

(iv) The generic minimum polynomial of x is integral.

An o-lattice M of ^ is an order if

(1) 1 € M.

(ii) x, y

e

M implies xU e M.

Clearly an order M of ^ is a Jordan algebra over o. If x € M then

o[x] C M. Hence o[x] is finitely generated and x is integral. M is said

to be maximal if it is maximal with respec t to inclusion. A maximal order M

is distinguished if M is a maximal lattic e of integral elements .

PROPOSITION 2. Every integral element x

c

^ can be embedded in an

order of ^.

PROOF. Let G be the o-algebra of transformations of % generated by

U r ... By (4) and (5) it is a commutative associativ e ring and is finitely gen -

erated as an o-module (by U ,, U . ., i, j degree of m (7\)). Let M, be

1 1 1 X 1

x x , x

an G-stabl e o -lattic e of ^ ( e . g . LG, L any o-lattice) . Since M U and

o[x]U are finitely generated o-modules there exist s a 7\

6

K, 7\ £ 0 such

1

that IV^Uj^ C A 2 M

1

and °[x]U C 7JM Let M

2

= TT1U . Then we

have