Remark

The prime ideals $p_i \in \mathrm{Ass}(M/N)$ which are not minimal elements of this set are sometimes called the immersed prime ideals associated with $M/N$; if $M/N$ is finitely generated, for $p_0 \in \mathrm{Ass}(M/N)$ to be immersed, it is necessary and sufficient that $V(p_0)$ be not an irreducible component of $\mathrm{Supp}(M/N)$ (Chapter II, § 4, no. 3, Corollary 2 to Proposition 14); if $(Q(p))_{p \in \mathrm{Ass}(M/N)}$ and $(Q'(p))_{p \in \mathrm{Ass}(M/N)}$ are two reduced primary decompositions of $N$ in $M$, it may be that $Q'(p_0) \neq Q(p_0)$ (Exercise 24 (c)); a canonical reduced primary decomposition of $N$ in $M$ may always be defined by imposing supplementary condition on the primary submodules which appear in it (Exercise 4).