Definition 2

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

Commutative Algebra, chapter IV, § 2. Primary decomposition

Let $A$ be a Noetherian ring, $M$ an $A$ -module and $N$ a submodule of $M$ . A finite family $(\mathscr{Q}_i)_{i \in I}$ of submodules of $M$ which are primary with respect to $M$ and such that $N = \bigcap_{i \in I} \mathscr{Q}_i$ is called a primary decomposition of $N$ in $M$ .

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

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