ARITHMETICS OF IORDAN ALGEBRAS

55

is a maximal # -stabl e order of C then M is maximal (since 01 c 0-orders

7 n

of C , M C M implies M» C M' C M' + 01 so M' + 01 = M' + 01 ; therefore

n'

M C J n (M1 + 01) = ^n (M' + 01) = M). To se e the converse, localize and

complete. If C splits then M maximal implies M maximal by

P P P

Corollary 1. If C is not split O l + M ' is maximal J -stabl e by (5), (6)

and (7).

q. e. d.

COROLLARY 6. Let C be a quaternion division algebra over K the

quotient field of a Dedekind ring o. Assume that every completed localiza -

tion of C splits or has no symmetric prime (in particular this is satisfied if

K is global). Let M be an order of P - U(C , K, J ), n = 3 or n even

4. Then M is maximal if and only if M = ^ n M' and M' is a maximal

J -stabl e order of C .

7 n

PROOF. If M ^ J n M ' and M' is maximal J -stabl e then M is

7

maximal by Proposition 2. To prove the converse it suffices to show it for all

/\ /\

completed localizations . If C splits this is Corollary 1. Assume C is a

division algebra. If n = 3 the lattice L of Theorem 3 must be 2i-modular

for otherwise we should have the exception (C ha s no symmetric prime by

assumption). Hence (6) or (7) cannot occur. If n is even, L is a sum of

(subnormal) planes and an even number of lines so (6) and (7) cannot appear

a s M1. Hence under the above hypothesis if M is maximal then M' is a

maximal J -stabl e order of C .

7 n

q . e . d.