Annihilator of Noetherian module has Noetherian quotient
Let be a finite generating set for . Consider a -module map from to given by:
The kernel of this map is precisely , so the quotient is a submodule of .
Since is Noetherian, is Noetherian, and hence is Noetherian (as it is a submodule of a Noetherian module). But being Noetherian as a -module is equivalent to being a Noetherian ring.