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