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
Previous Page Next Page