I show the following:
Let $R$ be a commutative ring with unity and $I \subseteq R$ an ideal. Prove: if $R/I$ is a Noetherian ring and $I/I^2$ is a finitely-generated $R$-module, then $R/I^n$ is a Noetherian ring.
My work: I show that $I/I^n$ and $R/I$ are Noetherian $R/I^n$-modules. I can show they are $R/I^n$-modules, but I cannot show they are Noetherian. E.g., why is $I/I^2$ a Noetherian $R/I^2$-module? It is only finitely-generated $R$-module, so it should be a finitely-generated $R/I^2$-module.