Let M be an Artinian A-module and let I be the annihilator of M in A. Is A/I necessarily an Artinian ring?
I believe the answer is no since this comes off of a similar result regarding Noetherian modules/rings whose proof required the fact that Noetherian modules are automatically finitely generated. However, I'm having a little trouble fully working out a counterexample.
A non-Noetherian Artinian module we saw in class seems like a good candidate, since non-Noetherian means it isn't finitely generated, but I'm not sure how to get at the annihilator of it. The module in question is specifically $M=\mathbb{Q}/\mathbb{Z}[p^{\infty}] = \{a/b \in \mathbb{Q} : p^n a/b \in \mathbb{Z}$ for some n$\}/\mathbb{Z}$ as a $\mathbb{Z}$-module (where p is a prime number), and I know (again from class) that every proper subgroup (i.e., submodule) of M is of the form $M[p^n]$, but I'm not entirely sure what the annihilator of M in $\mathbb{Z}$ looks like.
The $p$-primary subgroup of $\mathbf{Q}/\mathbf{Z}$ is an example of the type sought. It is an Artinian $\mathbf{Z}$-module, its annihilator is $0$ (no non-zero element of $\mathbf{Z}$ can kill every element of this group because $p^{-n}+\mathbf{Z}$ has order $p^n$), and $\mathbf{Z}=\mathbf{Z}/\{0\}$ is not Artinian.