Krull intersection theorem

Given a Noetherian ring $A$, an $A$-module $M$, and an ideal $I$ inside the radical of $A$, we have that $M$ is separated with respect to the $I$-adic topology.

Furthermore, if $A$ is also an integral domain and $J\subset A$ is a proper ideal, we have

${\displaystyle \bigcap _{n>0}}{J}^n=(0)$