Theorem 2 (Krull)

Tag 04G9, permanent. Cite it as [Bourbaki, Tag 04G9].

Commutative Algebra, chapter III, § 3. m-adic topologies on Noetherian rings

*Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$ , $E$ a finitely generated $A$ -module and $F$ a submodule of $E$ . Then the $m$ -adic topology on $F$ is induced by the $m$ -adic topology on $E$ .

It follows from no. 1, Proposition 1 that the filtration induced on $F$ by the $m$ -adic filtration on $E$ is m-good and the conclusion then follows from Proposition 4.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 3
Proposition 5, § 3
Proposition 7, § 3
Corollary 3, § 3
Proposition 1, § 5
Corollary, § 4

Cites

no. 1, Proposition 1
Proposition 4