Exercise 30, § 2

Let $M$ be an $A$-module, and let $N$ be a submodule of $M$ admitting a reduced primary decomposition $N = \bigcap_{i=1}^{n} Q_i$; let p_i be the prime ideal associated with $Q_i$.

a) Now prove that if $b$ is an ideal of $A$ which is not contained in any of the $p_i$, one has $N : b = N$ (cf. exerc. 17).

b) If each of the $Q_i$ is strongly primary in $M$, now prove conversely that if one has $N : b = N$, $b$ is not contained in any of the $p_i$.

(Notice that if $b \subset p_i$ and $P = \bigcap_{j \neq i} Q_j$, one has $b^r P \subset N$ for a suitable integer $r$, and deduce that if one had $N : b = N$, one would have $P = N$.) Deduce from this that there then exists $\beta \in b$ such that $N : (\beta) = N$.