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
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