Artin–Rees lemma

The set of Artin -Rees, named after Emil Artin and David Rees, is a set of commutative algebra. He makes a statement about products of powers of ideals of a Noetherian ring and finitely generated modules. The kit can be used to detect a certain topology of a sub-module as relative topology.

Wording of the sentence

It is an ideal in a commutative noetherian ring. Next are a finitely generated module and a submodule. Then there is a number such that for all:

Applications

Is any module as defined potencies

A base of neighborhoods in and thus a topology, called the -adic topology. In this a lot if and only open when there is a, for every with. In the situation above theorem so wear and the sub-module the -adic topology, contributes as a subset but also the relative topology of the -adic topology. With the help of the theorem of Artin -Rees it is no more difficult to show the equality of these two topologies.

The set of Artin -Rees can also be used to prove the average rate of Krull.