In a Noetherian ring, powers of a Jacobson-radical ideal have zero intersection.Suggest a slogan
Let \(R\) be a commutative Noetherian ring and let \(I\subseteq R\) be an ideal contained in the Jacobson radical \(J(R)\). Then \[ \bigcap_{n\ge0}I^n=0. \] In particular, \(\bigcap_{n\ge0}\mfm^n=0\) for every Noetherian local ring \((R,\mfm)\).
Let \(N=\bigcap_{n\ge0}I^n\). By the Artin--Rees lemma applied to \(N\subseteq R\), there is \(c\ge0\) with \(I^n\cap N=I^{n-c}(I^c\cap N)\) for all \(n\ge c\). Since \(N\subseteq I^n\) for every \(n\), taking \(n=c+1\) gives \(N=I(I^c\cap N)=IN\). The ideal \(N\) is finitely generated because \(R\) is Noetherian, so Nakayama's lemma with \(I\subseteq J(R)\) gives \(N=0\).
What is Krull's intersection theorem?