Corollary 6

Tag 05L9, permanent. Cite it as [Bourbaki, Tag 05L9].

Algebra, chapter II, § 1. Modules

Let M and N be two submodules of an A-module E such that E/M and E/N are modules of finite length; then E/(M \cap N) is of finite length and

\text{(41)} \quad \text{long}(E/(M \cap N)) + \text{long}(E/(M + N)) = \text{long}(E/M) + \text{long}(E/N).

It suffices to apply Corollary 3 to the exact sequence (30)

0 \to E/(M \cap N) \to (E/M) \oplus (E/N) \to E/(M + N) \to 0

using the fact that

\text{long}((E/M) \oplus (E/N)) = \text{long}(E/M) + \text{long}(E/N).

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

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