Suppose is a commutative ring with . We say that -module module is Noetherian if every ascending chain of submodules in eventually terminates. This is similar to how we define a ring to be Noetherian (a commutative ring is Noetherian if every ascending chain of ideals in eventually terminates).
In “Undergraduate Commutative Algebra” by Miles Reid, the following proposition is proved (page 53), from which some important properties of Noetherian modules can be derived.
Proposition. Let be a short exact sequence of -modules. Then,
I have written up proof of this here. (I have filled in some details, where Miles have left to the reader).
Here are some consequences:
(1) If are Noetherian modules (for ), then is Noetherian.
(2) If is a Noetherian ring, then -module is Noetherian if and only if it is finitely generated -module.
(3) If is a Noetherian ring, is a finitely generated -module, then any subdmoule is again finitely-generated -module.
Proof. (1) We recall that direct sum can be realized as an exact sequence , where , and . Applying the Proposition, we obtain that is Noetherian module. By simple induction, this is extended to direct sum of any finite number of Noetherian modules.
(2) If -module is finitely generated, then we have a surjection for some positive integer , where is a free module of rank . In other words, we have an exact sequence . From (1), we know is Noetherian. By the one of the directions () of the Proposition, we obtain that is also Noetherian. Conversely, if is a Noetherian -module, it is clear that is finitely generated -module, for otherwise we would obtain ascending chain of submodules
that does not terminate. Here is the submodule generated by elements .
(3) Since is finitely generated -module, we get from (2) that is a Noetherian -module. It is clear that any submodule of a Noetherian module is again Noetherian module. So a submodule is Noetherian. Applying (2) again, we get that is finitely-generated -module, as desired.