# Noetherian Rings and Ideals Associated with $\text{Ann}(M)$

If $M$ is a finitely generated $A$-module where $A$ is Noetherian and $I$ is an ideal of $A$ such that the support of $M$, $\mathrm{Supp}(M)$ is a subset of $V(I)$ the set of prime ideals containing $I$. How do i know if there exists an $n$ such that $I^n$ is a subset of $\mathrm{Ann}(M)$?

It is relevant that if $M$ is a finitely generated module then $Supp(M)=V(\text{Ann}(M))$.

• thank you john, i really need to start learning latex – MichaelMoore Sep 19 '15 at 20:20

Hint. $V(J)\subset V(I)\implies \mathrm{rad}(I) \subset \mathrm{rad}(J)$, so $I \subset \mathrm{rad}(J)$. Now use that $I$ is finitely generated.