A module 
is Noetherian if it obeys the ascending
chain condition with respect to inclusion, i.e., if every set of increasing sequences
of submodules eventually becomes constant.
If a module 
is Noetherian, then the following are equivalent.
1. 
satisfies the ascending chain condition
on submodules.
2. Every submodule of 
is finitely generated.
3. Every set of submodules of 
contains a maximal element.
