Corollary 2

Tag 05L5, permanent. Cite it as [Bourbaki, Tag 05L5].

Algebra, chapter II, § 1. Modules

Let $u : M \to N$ be an $A$ -module homomorphism. If $M$ or $N$ is of finite length, so is $\operatorname{Im}(u)$ . If $M$ is of finite length, so is $\operatorname{Ker}(u)$ and

\operatorname{long}(\operatorname{Im}(u)) + \operatorname{long}(\operatorname{Ker}(u)) = \operatorname{long}(M).

N

is of finite length, so is

\operatorname{Coker}(u)

and

\operatorname{long}(\operatorname{Im}(u)) + \operatorname{long}(\operatorname{Coker}(u)) = \operatorname{long}(N).

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

Cited by

Remark, § 1
Theorem 6, § 14

Cites

Nothing.