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
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