Let $M$ be a finite $R$-module that is noetherian and such that $M_{\mathfrak{p}}$ is artinian for each $\mathfrak{p}\in \text{Spec}(R)$. Then $M$ is an artinian $R$-module.
I have tried using a proof with Zorn's lemma: Let $\Gamma = \{ N: N\subseteq M \text{ is artinian } \}$. Since $0\in \Gamma$, and $M$ is noetherian, $\Gamma$ contains a maximal element, call it $N$. But then $N_{\mathfrak{p}}$ is a maximal artinian submodule of $M_{\mathfrak{p}}$ (I don't think this is true (a priori); I think proving this is equivalent to proving the wanted statement) for all $\mathfrak{p}$, so $N_{\mathfrak{p}} = M_{\mathfrak{p}}$ for all $\mathfrak{p}$, implying $N=M$.
I have also tried showing Jac$(R)M = \mathfrak{m}_1\cap \dots \cap \mathfrak{m}_t M$ where $\mathfrak{m}_t$ are finitely many maximal ideals, with which I have had no luck.