Definition 3

Let $M$ be a module over a Noetherian ring and $N$ a submodule of $M$. A primary decomposition $N = \bigcap_{i \in I} Q_i$ of $N$ in $M$ is called reduced if the following conditions are fulfilled:

(a) there exists no index $i \in I$ such that $\bigcap_{j \neq i} Q_j \subset Q_i;$

(b) if $\operatorname{Ass}(M/Q_i) = \{p_i\}$, the $p_i$ (i $\in I$) are distinct.

From every primary decomposition $N = \bigcap_{i \in I} Q_i$ of $N$ in $M$ a reduced primary decomposition of $M$ in $N$ can be deduced as follows: let $J$ be a minimal element of the set of subsets $I'$ of $I$ such that $N = \bigcap_{i \in I'} Q_i$. Clearly $(Q_i)_{i \in J}$ satisfies condition (a). Then let $\Phi$ be the set of $p_i$ for $i \in J$; for all $p \in \Phi$, let $H(p)$ be the set of $i \in J$ such that $p_i = p$ and let $Q(p) = \bigcap_{i \in H(p)} Q_i$; it follows from Proposition 2 of no. 1 that $Q(p)$ is $p$-primary with respect to $M$; further $N = \bigcap_{p \in \Phi} Q(p)$ and the family $Q((p))_{p \in \Phi}$ is therefore a reduced primary decomposition of $N$ in $M$.

We shall see that the primary decomposition defined in the proof of Theorem 1 of no. 2 is reduced; this follows from the following proposition: