I am stuck on this homework problem:
Let $R$ be a commutative ring. Suppose $M$ is a Noetherian $R$-module. Prove that $M$ is finitely generated and that $R / \textrm{Ann}_R (M)$ is Noetherian.
It is fairly trivial that $M$ is finitely generated; I am stuck on the second statement. The two characterisations of Noetherianity I am familiar with are
R is Noetherian if and only if all of its ideals are finitely generated
and the Ascending Chain Condition.
I tried to construct an ascending chain of ideals in $R / \textrm{Ann}_R (M)$ starting from an ascending chain of submodules in $M$ and then tried to write an element of the $(i+1)$-th ideal as a linear combination of the $(i)$-th ideal and tried to incorporate the fact that we divided by the annihilator, but this horribly failed and now my house is on fire, my dog is dead and my wife signed the divorce papers.
In particular, starting from a chain of submodules $$M_1 \subseteq M_2 \subseteq \cdots,$$ I defined the $i$-th ideal $I_i \subseteq R / Ann$ as the ideal which annihilates $M_i$, but then I get a descending chain...
I also went the other way, starting from a chain of ideals $$I_1 \subseteq I_2 \subseteq \ldots$$ and constructing $M_i := I_i(M)$.
Any help in the form of hints and pointers, not full answers would be appreciated! (I'd rather construct the full proof myself, but I've been stuck on this for a couple days now so I need some stepping stones)