Let $A$ be an integral domain and noetherian and let $I \subset A$ a proper ideal such $I*\cap_{n\geq0} I^n = \cap_{n\geq0} I^n$. Prove that $\cap_{n\geq0} I^n = (0)$
I'm trying to prove it by getting a contradiction taking a non-zero element from $\cap_{n\geq0} I^n$ but I'm getting nothing. I will appreciate if someone gives me a hint
Since $A$ is Noetherian, $J:=\bigcap_{n\geq 0} I^n$ is finitely generated as an ideal, and hence as an $A$-module. By Nakayama's lemma, there is an element $a\in A$ such that $a\equiv 1 \pmod{I}$ and $aJ=0$. Let $x\in J$, then $ax=0$ (because $ax\in aJ$) and since $a\neq 0$ and $A$ is an integral domain, we have $x=0$. Consequently $J=0$.