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
- Let . Then,
- There exists such that
- Krull intersection theorem for Jacobson radical, also covers the case of a local ring
- Krull intersection theorem for domains
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.
Now consider the filtration:
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.
- ''Dimensionstheorie in Stellenringen by Wolfgang Krull, 1938
- Book:Eisenbud, Page 152