Relation with other properties
- A ring is Noetherian as a module over itself if and only if it is a Noetherian ring. This is because ideals of a ring are precisely its submodules.
- Every finitely generated module over a Noetherian ring is a Noetherian module.
Any submodule of a Noetherian module is Noetherian.
If a module has a Noetherian submodule and the quotient module is Noetherian, the module itself is Noetherian.
Any quotient module of a Noetherian module is Noetherian.