ARITHMETICS OF JORDAN ALGEBRAS

23

refining Proposition 2. It should be noted that, by structure theory,

Proposition 2 and Corollary I. 2 complete the proof of Theorem I. 2.

§2. Maximal orders of matrix algebras .

Let & be a finite dimensional central division algebra over K.

Consider & the ring of n x n matrices with entries in $. Let it act on an

n

n-dimensional left & -vecto r spac e V. Let r be a maximal order of $, L an

D-lattice of V and E(L) = {A

€

& | LA C L} . Chevalley [5] has studied the

orders E(L) of $ . We will se e that E(L) is a maximal order of $ . The

n n

converse is als o true; that is , any maximal o-order of & = E(L) for some

O-lattice L, D a maximal order of $. Indeed let F be an o-order of &

' n

and, considering V as a K-vector space , let L be an F stable o-lattice

of lr (e. g. L = xF, x any non-zero vector of lr). Let

D = {a e $| aL C L }, D is clearly an o-order of & and is therefore con -

tained in a maximal order 0 of $. Let L = OL ; L is an O-lattice and

LF = (rLjJF = rfLjF) - tLl = L. Therefore F C E(L). If F is maximal then

F must equal E(L). It should be noted that O and a fortiori L are not

uniquely determined by F. We recall a few result s of Chevalley.

LEMMA 2. ([5], p. 11, 18) Let © be a fixed maximal order of &.

Any O-lattice L C 1/ can be written L = 31, x, 0 2J x © . . . 0 21 x , x, an

1 1 2 2 n n 1

arbitrary vector of V, 21. (fractional) left O-ideals . Hence E(L) is

isomorphic to