ARITHMETICS OF JORDAN ALGEBRAS 11

Also M' is an o-lattice of $ which contains 1. We wish to show that M'

is an order. Let a. € L, b. c M, i = 1, 2; (*a, + b . )U ,, =

i ' i ' 1 1 ota +b

« a U + *a,U

L

+ « a U

L

+ b , U + b U

u

+ b U . Since

1 « a

2

1 « a

2

, b

2

1 b

2

1 ora2 1 ara2, b

2

1 b

2

a a

0

€ 9{r'1\ U = 0 . Also b.U . c M and aa.U , € «(M n R^" 1 *) = «L.

2 a a

2

1 b

2

1 b

2

Let a' = a U , . Then a' € ^ ( r " 1 ' , so a' o a = 1U , e &(r*. There -

1 1 l , b

2

1 ' 1 2 a i * a 2

2 2 2

fore, by (2) and (3), a a ^

fa

= a { a ^ b ^ = a { b ^ a ^ =

« Z [(b

2

o a

x

) o a

2

- { a

l b 2

a

2

} ] = " V ^ ^ ) . a

2

- b ^ ^ ^ ] =

" 2 K ° a 2 " b 2 U a

r

a

2

1 = ° S i n c e a i ° a 2 a n d b 2 U a

r

a

2

€ * W = °-

Finally by (3) a b ^

b

= " { a ^ * ^ } = a[(a

2

0 ^ ) 0 ^ - { b ^ b ^ ]

€

«L

(r-1)

sinc e a

0

o h = a0LL , c M n ft and (a

n

o b_ ) © b =

Z 1 Z 1 , D .

Z 1 Z

(a0 o bjCJ, , c M n / "

1

' = L.

2 1 l , b

2

q . e . d .

We consider next th e converse, namely: if $ is semisimple does it

contain maximal orders? Suppose # is simple then its centroid T is a finite

field extension of K. In general £ th e integral closure of 0 in r need

not be a finitely generated o-module (see for example [42] p . 443). The same

problem is encountered in classica l associativ e arithmetic where one assume s

either that © is a finitely generated o-module (this holds if r/ K is separable)

or that the algebra is central .