Page 150 - Advanced Linear Algebra
P. 150
134 Advanced Linear Algebra
:~ :
:
is easily seen to be a submodule of 4 . Hence, is finitely generated, say
: ~ ºº" Á à Á " »». Since " :, there exists an index such that " : .
Therefore, if ~ max ¸ ÁÃÁ ¹ , we have
¸" ÁÃÁ" ¹ :
and so
: ~ º º " Á Ã Á " » » : : : b Ä b :
which shows that the chain 5.1 is eventually constant.
(
)
For the converse, suppose that 4 satisfies the ACC on submodules and let be
:
a submodule of 4 . Pick " : and consider the submodule : ~ º º " » » :
:
generated by " . If : ~: , then is finitely generated. If : £: , then there is
:
a " : c : . Now let : ~ºº" ," »» . If : ~: , then is finitely generated.
If :£ : , then pick " : c : 3 and consider the submodule
: ~ ºº" " " »».
,,
3
3
Continuing in this way, we get an ascending chain of submodules
ºº"»» ºº"Á "»» ºº"Á "Á "»» Ä :
If none of these submodules were equal to , we would have an infinite
:
ascending chain of submodules, each properly contained in the next, which
contradicts the fact that 4 satisfies the ACC on submodules. Hence,
:~ ºº" Á à Á " »» for some and so : is finitely generated.
Our goal is to find conditions under which all finitely generated -modules are
9
Noetherian. The very pleasing answer is that all finitely generated -modules
9
are Noetherian if and only if is Noetherian as an -module, or equivalently,
9
9
as a ring.
Theorem 5.8 Let be a commutative ring with identity.
9
1 9 ) is Noetherian if and only if every finitely generated 9 -module is
Noetherian.
)
2 Let be a principal ideal domain. If an -module 4 is -generated, then
9
9
any submodule of 4 is also -generated.
)
Proof. For part 1 , one direction is evident. Assume that is Noetherian and
9
let 4~ ºº" Á à Á " »» be a finitely generated 9 -module. Consider the
epimorphism ¢9 ¦ 4 defined by
² ÁÃÁ ³ ~ " b Ä b "
Let be a submodule of 4 . Then
:
