# Krull intersection theorem for modules

This article defines a result where the base ring (or one or more of the rings involved) is Noetherian

View more results involving Noetherianness or Read a survey article on applying Noetherianness

This fact is an application of the following pivotal fact/result/idea:Artin-Rees lemma

View other applications of Artin-Rees lemma OR Read a survey article on applying Artin-Rees lemma

This fact is an application of the following pivotal fact/result/idea:Cayley-Hamilton theorem

View other applications of Cayley-Hamilton theorem OR Read a survey article on applying Cayley-Hamilton theorem

## Contents

## Statement

Let be a Noetherian ring and be an ideal inside . Suppose is a finitely generated module over . Then, we have the following:

- Let . Then,
- There exists such that

## Results used

## Applications

- Krull intersection theorem for Jacobson radical, also covers the case of a local ring
- Krull intersection theorem for domains

## Proof

### The intersection equals its product with

We first show that the intersection equals its product with . This is the step where we se the Artin-Rees lemma.

Let:

Now consider the filtration:

this is an -adic filtration and the underlying ring is Noetherian, hence by the Artin-Rees lemma, the following filtration is also -adic:

Since each contains , the filtration below is the same as the filtration:

This being -adic forces that .

### Finding the element

Since , we can find an element such that . This is an application of the Cayley-Hamilton theorem: we first find the Cayley-Hamilton polynomial, then observe that is a root of the polynomial, and then take the negative of the sum of all coefficients of higher degree terms.

## References

- ''
*Dimensionstheorie in Stellenringen*by Wolfgang Krull, 1938

### Textbook references

- Book:Eisenbud, Page 152