Example 4

Tag 0005, permanent. Cite it as [Bourbaki, Tag 0005].

Algebra, VIII, p. 2, § 1. Artinian Modules and Noetherian Modules

Let $p$ be a prime number and $M_p$ the $p$ -primary component of the torsion $\mathbf{Z}$ -module $\mathbf{Q}/\mathbf{Z}$ (VII, §2, No. 2, p. 7). Every submodule of $M_p$ is equal to either $M_p$ or $p^{-n}\mathbf{Z}/\mathbf{Z}$ for an integer $n\in \mathbf{N}$ (VII, §2, p. 54, Exercise 3). Consequently, $M_p$ is an Artinian but not Noetherian $\mathbf{Z}$ -module.

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

VII, §2, No. 2, p. 7
VII, §2, p. 54, Exercise 3