Corollary

Let $A$ be a Noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$ and $(Q_i)_{i \in I}$ a primary decomposition of $N$ in $M$. Then $\mathrm{Card}(I) \geq \mathrm{Card}(\mathrm{Ass}(M/N))$; for $(Q_i)_{i \in I}$ to be a reduced primary decomposition, it is necessary and sufficient that $\mathrm{Card}(I) = \mathrm{Card}(\mathrm{Ass}(M/N))$.

It follows from the remarks preceding Proposition 4 that there exists a reduced primary decomposition $(R_j)_{j \in J}$ of $N$ in $M$ such that $\mathrm{Card}(J) \leq \mathrm{Card}(I)$; the first assertion then follows from the second and the latter is a consequence of Proposition 4.