Exercise 18, § 2

(a) Let $A$ be a ring and $M$ an $A$-module. Show that, if $p$ is a minimal element of $\operatorname{Ass}_f(M)$, the saturation of $\{0\}$ with respect to $p$ in $M$ is $p$-primary in $M$ (cf. § 1, Exercise 17 (b)).

(b) Let $p$ be a prime ideal of $A$. Show that for every integer $n > 0$ the saturation in $A$ of $p^n$ with respect to $p$ is $p$-primary in $A$ (use Exercise 14 (a)); this saturation is denoted by $p^{(n)}$ and is called the $n$-th symbolic power of $p$.

(c) Show that, if every prime ideal of $A$ is maximal, then, for every $A$-module $M$ and every submodule $N$ of $M$, $N$ is the intersection of a family (finite or otherwise) of primary submodules in $M$ (use (a) and Exercise 17 (i) of § 1).