Corollary

Tag 000R, permanent. Cite it as [Bourbaki, Tag 000R].

Algebra, VIII, p. 9, § 1. Artinian Modules and Noetherian Modules

Let A be a commutative ring.

a) Let M be a Noetherian A-module. The ring $A_M$ is Noetherian.

b) Let M be an A-module of finite length. The ring $A_M$ is Artinian.

Let M be an A-module. Under the assumptions of a) or b), the A-module M is finitely generated. Since A is commutative, $A_M$ is contained in the ring End $_A$ (M), so that the countermodule of M is finitely generated. It then suffices to apply Proposition 6.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 6