In a Noetherian domain $R$, if $I\neq R$ then $\bigcap_{n=1}^\infty I^n=0$?

Question

I am studying the Krull's Intersection Theorem:

Let $I$ be an ideal in a Noetherian ring $R$. Then $\bigcap_{n=1}^\infty I^n=0$ if and only if $1+I$ contains no zero-divisors.

I am not clear how to prove that in a Noetherian domain $R$, if $I\neq R$ then $\bigcap_{n=1}^\infty I^n=0$. I think since $I\neq R$, then $I$ has no units and hence every element of $I$ is contained in a maximal ideal $M$ of $R$. How can I prove that $1+I$ has no zero-divisors?

Answer 1

Since $R$ is a domain, the only zero-divisor in $R$ is $0$. So you just have to show $0\not\in 1+I$. I think you can finish the argument from there.