Noetherian module

Symbol-free definition
A module over a commutative unital ring is termed Noetherian if every submodule of it is finitely generated.

Weaker properties

 * Finitely generated module

Incomparable properties

 * Artinian module

Facts

 * 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.

Metaproperties
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.