Krulls intersection theorem without Noetherian

Question

Krulls intersection theorem states

Let $R$ be a noetherian integral domain and $I\subset R$ an proper ideal. Then $\cap_{n>0}I^n=0$.

What are some simple counterexamples if we forget the fact that $R$ is noetherian or an integral domain?

Answer 1

How about $R$ is the ring of $C^\infty$ maps from $\Bbb R$ to $\Bbb R$ and $I$ be those elements of $R$ with $f(0)=0$? Then a function with vanishing Taylor series at the origin will lie in $\bigcap_n I^n$.

Answer 2

$\mathbb{Z}/6\mathbb{Z}$ is noetherian but not an integral domain. Take the proper ideal $I=\{0, 2, 4\}$.