Corollary 5

Tag 05L8, permanent. Cite it as [Bourbaki, Tag 05L8].

Algebra, chapter II, § 1. Modules

Let $M$ be an $A$ -module the sum of a family $(N_i)$ of submodules of finite length. Then $M$ is of finite length and

\operatorname{long}(M) \leq \sum_i \operatorname{long}(N_i).

Moreover, for the two sides of (40) to be equal, it is necessary and sufficient that M be the direct sum of the $N_i$ .

It has been seen (no. 7, formula (28)) that there is a canonical surjective linear mapping $h : \bigoplus_i N_i \to M$ ; hence the corollary follows from (35).

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

Cited by

Corollary 5, § 7
Proposition 10, § 4

Cites

Nothing.