If $R$ noetherian ring, $I \subset R$ proper ideal such that every element of $1+I$ is invertible. Show that: $$\bigcap_{n>0} I^n = (0)$$
Idea: We know that $I \subset J(R)$, with $J(R)$ the Jacobson's ideal of $R$, we want use Nakayama's Lemma
Nakayama's Lemma: If $M$ a finitely generated $R$-module and $I$ an ideal of $R$. If $I \subset J(R)$ and $IM=M$, then $M=(0)$
in our case $I=I$, $M=\bigcap I^n$ is finitely generated because $R$ noetherian, we can't prove that $I (\bigcap I^n) = \bigcap I^n$
