Corollary 4

Tag 05L7, permanent. Cite it as [Bourbaki, Tag 05L7].

Algebra, chapter II, § 1. Modules

Let $M$ and $N$ be two submodules of finite length of an $A$ -module $E$ ; then $M + N$ is of finite length and

\operatorname{long}(M + N) + \operatorname{long}(M \cap N) = \operatorname{long}(M) + \operatorname{long}(N).

It suffices to apply Corollary 3 to the exact sequence (29) (no. 7).

0 \to M \cap N \to M \oplus N \to M + N \to 0

using the fact that

\operatorname{long}(M \oplus N) = \operatorname{long}(M) + \operatorname{long}(N)

by (34).

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

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