Let \(\mathrm{I} \in \operatorname{Id}(R)\) with \(R\) Noetherian and \(M\in {}_{R}{\mathsf{Mod}} ^{\mathrm{fg}}\) with \(N\leq_{ {}_{R}{\mathsf{Mod}} } M\) a submodule.
Then there exists an integer \(k \geq 1\) so that, for \(n \geq k\), \begin{align*} I^{n} M \cap N=I^{n-k}\left(\left(I^{k} M\right) \cap N\right) . \end{align*}
- Used to prove the Krull’s intersection theorem: this is a separable topology iff \(1+I\) contains no zero divisors,which holds e.g. if \(I \subseteq {J ({R}) }\) (the Jacobson radical).
- Used to prove that adic completion is exact, and \(M{ {}_{ \widehat{I} } } \cong M \otimes_R R{ {}_{ \widehat{I} } }\)
- Topological interpretation: the \(I{\hbox{-}}\)adic topology on \(N\) is induced by the \(I{\hbox{-}}\)adic topology on \(M\).