since is Noetherian, is finitely generated. let and it's easy to see that

if is generated by as module, then is obviously generated by as module. this is true for any ideal of any ring

2. Let be a Noetherian ring and let Prove that is a finitely generated -module iff is a finitely generated -module.

conversely: since is Noetherian, is nilpotent, i.e. for some now is finitely generated module because it's finitely generated module.

applying part 1) of the problem gives us that and are both finitely generated (and hence Noetherian) module. thus is also Noetherian.

again and are Noetherian and so is Noetherian too. continuing this we'll get that is Noetherian (and so finitely generated) module.

(recall that a finitely generated module over a Noetherian ring is Noetherian and a module is Noetherian iff for any submodule of both and are Noetherian.)