24

M . L. RACINE

O,

2 J ~2 V

.. . 8 1 " ^

1 I n

\ \

©n . . . 81 '81

2 2 n

81 81,

n 1

where r. is the right order of 81, and by this "matrix" one means the set of

I I

matrices whose i j t h entries belong to the ideal in the i j t h place in the above

matrix.

We wish to show that E(L) is a maximal order. Assume the E(L) is

given as in the above "matrix". If E(L) is not maximal then E(L) + oA gen -

erates an order F for some A e $ , A 4 E(L). Since the idempotents e..,

n u

1 i n belong to E(L) we may assum e A = ae.. , a ? ^, for some matrix

unit e.. and a I 81. 81. the ijt*1 entry of the above "matrix". Since

U i J

(81."" 81.)" = 81," 81. the j i t h entry of the above "matrix", a8i.~ 8l.e.. C F and

i J l i } i i i

a8I." 81. CjD,. Let b € a8l, 81., b i JO. . Now D. is a maximal order therefore

J 1 T 1 J 1 1 1

the ring generated by D. + ob is not a finitely generated module, a contradiction.

LEMMA 3. ([5], p. 12, 18) Any ©-lattic e L can be written

L = 0yx 0 0y

2

Cy , 0 81y for some {y.} a bas e of V and some

n-1 n

I

(fractional) left O-idea l 81. Hence E(L) is isomorphic to