ARITHMETICS OF JORDAN ALGEBRAS 3 5

LEMMA 11 . Let E. = A~l& A., i = 1, 2, A.

€

C . Then the number of

l

I

n

I

i n ,

n- 1

maximal orders containing the order F = E_ n En is equal to II (r. ,. - r. +1)

1 Z . . l+l i

r l r n -1 1 =

where {p , . . . , p } are the elementary divisors of A A .

PROOF. We have A.FA71 = C n ((A A" 1 )" 1 © (A-A?1)). Applying

1 1 n 2 1 ' n 2 1

Lemma 9 to A^A, and absorbing one of the units of © into 0 , there

2 1 n n

exist s a unit U of fl such that u " A, FA7 U = C n D~ © D,

n l i n n

n r,

D = ) p e

t l

, r_ r_ . . . r . So F is isomorphic to

y n 1 — 2 — — n

r.-r. r.-r,

Y ? J l e.. + Y Oe.. . Assume F = Y y ] l

e

. , + Y C e . . . Then if

iJ iJ iJ iJ

E = A © A contains F, by Lemma 10 we may tak e A diagonal, say

must have s, s_ . . . s

1 — 2 — — n

n

n s. s , - s .

A = Y p 1 e . . . Since E = Y p J l e . . we

i=l i, J

(otherwise F (£ E) and F C E if and only if r. - r, s, - s, 0,

1 i n.

q. e. d.

COROLLARY 2. Let F = E, n E

0

, E. maximal orders of & . Then

1 2

I

n

F C E maximal implies E

€

{ E , E }, if and only if r - r 1.

\. Ct n i

LEMMA 12. Let E,, 1 = 1 , 2 , 3 , 4 be maximal orders of & . If

i n

rrri

Ex n E

2

= E3 n E

4

then { E j . E ^ = {E

3

,E

4

.

PROOF. We may assum e a s above that E, = n , E0 = Y $ • " *e .,

1 n 2 .L, T i j '

r.. r . . . . r , and E. n E = Y p J V . + Y Oe.. . By Lemma 10,

1 " 2 - - n 1 2 .L ^

13

^

13

s.-s. t.-t. - '

E = I ?

J L

e.., E = [ p

J

* e.. and E n E = E n E if and only if

max(s. - s., t. - t.) = r. - r. for i j and max(s. - s.,t . - t.) = 0, i j . We

J i J i J i j i 3 i —