Corollary 7

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

Algebra, chapter II, § 1. Modules

Let $(M_i)$ be a finite family of submodules of an A-module E such that the $E/M_i$ are modules of finite length. Then $E/(\bigcap_i M_i)$ is of finite length and

\text{(42)} \quad \text{long}\left(E/(\bigcap_i M_i)\right) \leq \sum_i \text{long}(E/M_i).

It has been seen (formula (27)) that there is a canonical injective linear mapping $E/(\bigcap_i M_i) \to \bigoplus_i (E/M_i)$ .

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

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