Corollary 2
Tag 072E, permanent. Cite it as [Bourbaki, Tag 072E].
Algebra, chapter VII, § 2. Torsion modules over a principal ideal domain
— The submodule N of the torsion A-module M is a direct factor if and only if N, is a direct factor of M, for every irreducible element π of A.
Indeed, if N and N' are two submodules of M, then M = N ⊕ N' if and only if M, − N, ⊕ N'_π for every irreducible element π of (Cor. 1).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.