Page 149 - Advanced Linear Algebra

Modules II: Free and Noetherian Modules 133

4 : finitely generated,  4 ¬ : finitely generated
Example 4.2 shows that this is not always the case and leads us to search for
conditions on the ring that will guarantee this property for -modules.
9
9
Definition An -module 4 is said to satisfy the ascending chain condition
9
(abbreviated ACC ) on submodules if every ascending sequence of submodules
:‹ :‹ :‹ Ä

of 4 is eventually constant, that is, there exists an index for which

:~ : b ~ : b k ~ Ä

Modules with the ascending chain condition on submodules are also called

(
Noetherian modules after Emmy Noether, one of the pioneers of module
)
theory .
Since a ring is a module over itself and since the submodules of the module 9
9
are precisely the ideals of the ring , the preceding definition can be formulated
9
for rings as follows.
Definition A ring 9 is said to satisfy the ascending chain condition
(abbreviated ACC ) on ideals if any ascending sequence

? ? ? ‹ ‹ ‹ Ä
of ideals of is eventually constant, that is, there exists an index for which

9
? ? ? b ~ ~ b 2 ~ Ä
A ring that satisfies the ascending chain condition on ideals is called a
Noetherian ring.

The following theorem describes the relevance of this to the present discussion.

Theorem 5.7
)
9
1 An -module 4 is Noetherian if and only if every submodule of 4 is
finitely generated.
2 In particular, a ring is Noetherian if and only if every ideal of is
)
9
9
finitely generated.
Proof. Suppose that all submodules of 4 are finitely generated and that 4
contains an infinite ascending sequence
:‹ :‹ :‹ Ä (5.1 )
3

of submodules. Then the union