Exercise 20, § 2

The notion of primary decomposition and that of reduced primary decomposition may be defined for any ring $A$ as in nos. 2 and 3 (using the notion of primary submodule defined in Exercise 12).

(a) Generalize Propositions 4 and 6 by replacing Ass by Ass, throughout.

(b) Let $E$ be a finitely generated $A$-module and $F$ a submodule of $E$. For every multiplicative subset $S$ of $A$ let $\operatorname{sat}_S(F)$ denote the saturation of $F$ in $E$ with respect to $S$. Show that, if $F$ admits a primary decomposition in $E$, every saturation of $F$ in $E$ is of the form $F : (a)$ for some $a \in A$. (Let $p_i$ ($1 \leq i \leq r$) be the elements of $\operatorname{Ass}_f(E/F)$. Show first that, using (a), that for every multiplicative subset $S$ of $A$ there exists $b \in A$ such that, if $T$ is the multiplicative subset of the $b^n$ ($n \geq 0$), then $\operatorname{sat}_T(F) = \operatorname{sat}_S(F)$, taking $b$ such that $b \in p_i$ if $p_i \cap S \neq 0$ and $b \notin p_i$ otherwise. Then prove that we may take $a = b^m$ for $m$ sufficiently large.)

(c) Give an example of a $\mathbf{Z}$-module $E$ such that $\{0\}$ is primary with respect to $E$ but there exist saturations $\operatorname{sat}_S(0)$ which are not of the form $0$: (a) (cf. Algebra, Chapter VII, § 2, Exercise 3).