Corollary 2

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

Commutative Algebra, chapter V, § 1. Notion of an integral element

If $A$ is a principal ideal domain, $A'$ is a free $A$ -module of rank $n$ . Every submodule of a free $A$ -module is then free (Algebra, Chapter VII, § 3, Theorem 1).

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

Cited by

Corollary 3, § 1
Corollary 1, § 8

Cites

*Algebra*, Chapter VII, § 3, Theorem 1