Page 167 - Advanced Linear Algebra
P. 167
Modules Over a Principal Ideal Domain 151
ºº »» # ºº »» l # ºº »» # Ä
must terminate since 4 is Noetherian and so there is an integer for which
(
eventually : ~ ¸ ¹ , giving 6.1 . )
Let # ~ # . The direct sum 4 ~ ºº#»» l ¸ ¹ clearly exists. Suppose that the
direct sum
4~ ºº#»» l :
exists. We claim that if 4 4 , then it is possible to find a submodule : b
also
for which : : b and for which the direct sum 4 b ~ ºº#»» l : b
exists. This process must also stop after a finite number of steps, giving
4~ ºº#»» l : as desired.
If 4 4 and " 4 ± 4 let
: ~ b º º : Á " c # » »
for 9 . Then : : b since " ¤4 . We wish to show that for some
9, the direct sum
ºº#»» l : b
exists, that is,
% ºº#»» q ºº: Á " c #»» ¬ % ~
Now, there exist scalars and for which
%~ # ~ b ²" c #³
and so if we find a scalar for which
for :
²" c #³ : (6.2)
then ºº#»» q : ~ ¸ ¹ implies that % ~ and the proof of existence will be
complete.
Solving for " gives
" ~ ² b ³# c ºº#»» l : ~ 4
so let us consider the ideal of all such scalars:
? ~¸ 9 " 4 ¹
?
Since ? and is principal, we have
? ~º »
for some . Also, since " ¤ 4 implies that ¤ ? .
