Let $A$ be a commutative noetherian ring with unity, $I$ is an ideal contained in the radical of $A$, how to show $\bigcap\limits_{n=0}^\infty I^n=0$?
I want to show $I \bigcap\limits_{n=0}^\infty I^n= \bigcap\limits_{n=0}^\infty I^n$, but I don't know how to show $ \bigcap\limits_{n=0}^\infty I^n\subset I \bigcap\limits_{n=0}^\infty I^n$.
Use Krull's intersection theorem to show that $IN =N$ (as $A$-modules), where $N$ is the intersection of the powers of $I$. Then your claim follows from Nakayama's lemma.
In case that you maybe only know the Artin-Rees lemma, consider the filtration given by the powers of $I$ and apply Artin-Rees. This will lead to the intersection theorem (if you do it quite generally) and thus to your equality.
