Let $\mathrm{I} \in \Id(R)$ with $R$ Noetherian and $M\in \rmod^\fg$ with $N\leq_{\rmod} M$ a submodule. 

Then there exists an integer $k \geq 1$ so that, for $n \geq k$,
$$
I^{n} M \cap N=I^{n-k}\left(\left(I^{k} M\right) \cap N\right) .
$$

- Used to prove the [Krull's intersection theorem](Unsorted/Krull's%20intersection%20theorem.md):  this is a [separable topology](separable%20topology) iff $1+I$ contains no zero divisors,which holds e.g. if $I \subseteq \jacobsonrad{R}$ (the [Jacobson radical](Jacobson%20radical)).
- Used to prove that [adic completion](Unsorted/adic%20completion.md) is exact, and $M\complete{I} \cong M \tensor_R R\complete{I}$
- Topological interpretation:  the $I\dash$adic topology on $N$ is induced by the $I\dash$adic topology on $M$.