Corollary 3

Tag 072F, permanent. Cite it as [Bourbaki, Tag 072F].

Algebra, chapter VII, § 2. Torsion modules over a principal ideal domain

— Let N be a submodule of the torsion A-module M. If, for every irreducible element π of A, either N, = 0 or $(M/N)_\pi = 0$ , then N is a direct factor of M.

Indeed, the condition $(M/N)_\pi = 0$ implies $N_\pi = M_\pi$ and Cor. 2 applies.

An A-module M is called semi-simple if every submodule of M is a direct factor (cf. A, VIII, § 3).

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

Cited by

Lemma 4, § 2

Cites

Cor. 2
VIII, § 3