Corollary 2

Tag 04JF, permanent. Cite it as [Bourbaki, Tag 04JF].

Let $M$ be an A-module and $(Q_i)_{i \in I}$ a finite family of submodules of $M$ . If $\bigcap_{i \in I} Q_i = \{0\}$ , then

\operatorname{Ass}(M) \subset \bigcup_{i \in I} \operatorname{Ass}(M/Q_i).

The canonical mapping

M \to \bigoplus_{i \in I} (M/Q_i)

is injective; then it suffices to apply Proposition 3 and its Corollary 1.

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

Cited by