Page 151 - Advanced Linear Algebra

Modules II: Free and Noetherian Modules 135

c ²:³~¸"9 “ ":¹

is a submodule of 9 and ² c : ³ ~ : . If every submodule of 9 is finitely
generated, then c ²:³ is finitely generated and so c ²:³ ~ º º# ÁÃÁ# . » »
Then is finitely generated by ¸ # Á Ã Á # ¹ . Thus, it is sufficient to prove the
:
theorem for 9 , which we do by induction on .

9
If ~ , any submodule of is an ideal of , which is finitely generated by
9
assumption. Assume that every submodule of 9 is finitely generated for all
  and let : be a submodule of 9 .

:
If € , we can extract from something that is isomorphic to an ideal of 9
be the “last coordinates” in
and so will be finitely generated. In particular, let :
:, specifically, let
Á ³  : for some

: ~ ¸² Á à Á Á ³ “ ² Á à Á c Á à Á c  9¹
:
9
The set is isomorphic to an ideal of and is therefore finitely generated, say
: ~ ºº »», where ~ ¸ Á à Á ¹ is a finite subset of : .
=
=

Also, let
Á ³ for some

: ~ ¸#  : “ # ~ ² Á à Á c Á à Á c  9¹

: :
be the set of all elements of that have last coordinate equal to . Note that
is a submodule of 9 and is isomorphic to a submodule of 9 c . Hence, the
inductive hypothesis implies that : is finitely generated, say : ~ º º = » » , where
= is a finite subset of .:
:  = has the form
By definition of , each
~ ² ÁÃÁ Á ³

Á

for Á  9 where there is a  : of the form
Á
Á
~ ² ÁÃÁ Á c Á ³

Let ~¸ Á Ã Á ¹ . We claim that is generated by the finite set r = .
=
:
=

To see this, let # ~ ² ÁÃÁ ³  : . Then ² Á ÃÁ Á ³  : and so

² ÁÃÁ Á ³ ~

~
for  9 . Consider now the sum