Lemma 1

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

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

Let A be a commutative Noetherian ring and m an ideal of A. Then the subring $A' = \sum_{n \geq 0} m^n X^n$ of $A[X]$ is Noetherian.

$A'$ is an A-algebra generated by $mX$ ; as A is Noetherian, $mX$ is a finitely generated A-module and the conclusion then follows from § 2, no. 10, Corollary 3 to Theorem 2.

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

Cited by

Proposition 1, § 3

Cites

§ 2, no. 10, Corollary 3 to Theorem 2