The condition above is generally hard to check, as we would need to check it for each submodule.
Instead we have the following proposition:
Proposition: Let be an -module with submodule .
Then is Noetherian if and only if and are Noetherian.
This means that every finite direct sum of Noetherian modules is Noetherian
Relation to finitely generated
Let be a Noetherian ring.
An -module is Noetherian if and only if it is finitely generated.
Proof
We need the following lemma:
Lemma: A ring is Noetherian if and only if is Noetherian as an -module.
Proof of Lemma: The submodules of are ideals of as a ring.
Let be Noetherian, then by definition the submodule being the whole module is finitely generated.
Next, let be finitely generated.
Thus, is a quotient of some free R-module.
Since is a Noetherian ring, then by the lemma then is a Noetherian -module and thus, is also Noetherian.
By the quotient property above we have that is Noetherian.