3 4
M. L. RACINE
The p l are called the elementary divisors of A and are uniquely determined.
PROOF. ([14], Chapter III, §7)
LEMMA 10. Let A

& be invertible and A = UB a s in Lemma 8. If
n
n
Y ©e,, c A" © A then b. , = 0, 1 i j n.
1=1 li n u ' - -
PROOF. We have A - 1 © A = B ^ u " 1 © UB = B " 1 © B. But B = DC with
n n n
D = Y p l e . . , C = 1 + y c. .e.. where c.. = p 1 b . . ; e.
t
c " D~ © DC or
ik
U
ij
13 13 U
ij' ii
n
C e . . C D O D. Induct first on the column index of the first row and then
n n
on the number of rows. Assume we have c = c =...= c = 0,
1 j n.
0 c
U
C e . . C
-1
0 - c
U
r.-r,
= a matrix with c in the lj position. Therefore p b c p J SO a n d
r
j
b . . p O. But v(b. .) r. or b . = 0, so b , must be 0. Complete the
! j l j 3 l j l j
proof by inducting on the number of rows.
q. e. d.
Note that since © is a principal ideal ring Lemma 3 implies that any maximal
order of & has the form A © A and conversely.
n n
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