Let $R$ be a Noetherian ring , $M$ a finitely generated $R$-module and let $J$ be an ideal such that $Supp(M) \subset V(J)$ where $V(J) = \{P \in Spec(R) : P \supseteq J\}$. How to show there exists some $k>0$ such that $J^{k}M=0$?
I know that when $M$ is finitely generated we have $Supp(M)=V(ann(M))$ but I still don't see it. Can you please help?
EDIT:
So far I've figured it out that $J \subseteq \sqrt{Ann(M)}$. Now since $R$ is Noetherian then $J$ is finitely generated, say by $b_{1},b_{2},...,b_{t}$. For each $b_{i} \in \langle b_{1},..,b_{t} \rangle$ we have $b_{i}^{n_{i}} M =0$ for some natural $n_{i}$. Can we simply take then $k=n_{1} + n_{2}+...+n_{t}$ then by the binomial theorem $J^{k}M=0$?