## Tag `00IP`

Chapter 10: Commutative Algebra > Section 10.50: More Noetherian rings

Lemma 10.50.4 (Krull's intersection theorem). Let $R$ be a Noetherian local ring. Let $I \subset R$ be a proper ideal. Let $M$ be a finite $R$-module. Then $\bigcap_{n \geq 0} I^nM = 0$.

Proof.Let $N = \bigcap_{n \geq 0} I^nM$. Then $N = I^nM \cap N$ for all $n \geq 0$. By the Artin-Rees Lemma 10.50.2 we see that $N = I^nM \cap N \subset IN$ for some suitably large $n$. By Nakayama's Lemma 10.19.1 we see that $N = 0$. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 11631–11636 (see updates for more information).

```
\begin{lemma}[Krull's intersection theorem]
\label{lemma-intersect-powers-ideal-module-zero}
Let $R$ be a Noetherian local ring. Let $I \subset R$ be
a proper ideal. Let $M$ be a finite $R$-module.
Then $\bigcap_{n \geq 0} I^nM = 0$.
\end{lemma}
\begin{proof}
Let $N = \bigcap_{n \geq 0} I^nM$.
Then $N = I^nM \cap N$ for all $n \geq 0$.
By the Artin-Rees Lemma \ref{lemma-Artin-Rees}
we see that $N = I^nM \cap N \subset IN$ for
some suitably large $n$. By Nakayama's Lemma \ref{lemma-NAK}
we see that $N = 0$.
\end{proof}
```

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