Page 158 - Advanced Linear Algebra
P. 158
142 Advanced Linear Algebra
~ #~ " ~ 6 7 "
Since ²" ³ and ° are relatively prime, the order of ² ° ³" is equal to
²" ³ ~ ²#³ ~ , which contradicts the equation above. Hence, .
It is clear that ºº" b Ä b " »» ºº" »» l Ä l ºº" »» . For the reverse
inclusion, since and ° are relatively prime, there exist Á 9 for which
b ~
Hence
"~ 6 b 7 "~ "~ ²" b Ä b " ³ ºº" b Ä b " »»
Similarly, " ºº" b Ä b " »» for all and so we get the reverse inclusion.
Finally, to see that the sum above is direct, note that if
#b Ä b # ~
, then each must be , for otherwise the order of the sum on the
where # ( #
left would be different from .
) ~° 9
For part 2 , the scalars are relatively prime and so there exist
for which
b Ä b ~
Hence,
# ~ ² bÄb ³# ~ #bÄb #
#³ ~ °gcd ² Á and are relatively prime,
Since ² ³ ~ and since
we have ² #³ ~ . The second statement follows from part 1 . )
Free Modules over a Principal Ideal Domain
We have seen that a submodule of a free module need not be free: The
submodule d ¸ ¹ of the module d { over itself is not free. However, if 9
{
{
is a principal ideal domain this cannot happen.
Theorem 6.5 Let 4 be a free module over a principal ideal domain . Then
9
rk
any submodule of 4 is also free and ²:³ rk ²4 . ³
:
Proof. We will give the proof first for modules of finite rank and then
generalize to modules of arbitrary rank. Since 4 9 where ~ rk ²4³ is
finite, we may in fact assume that 4 ~9 . For each , let
