Exercise 4, § 3

Let $E$ be a torsion-free module over a principal ideal domain $A$. a) Show that, if $F$ is a pure submodule of $E$ (VII, p. 55, Ex. 7), then $E/F$ is torsion-free, and conversely. b) Let $F$ be a submodule of $E$ generated by a maximal pseudofree family (VII, p. 55, Ex. 4); then $F$ is a pure, free submodule of $E$. Show that, for all $1 \in E/F$, there exist a noninvertible element $a$ of $A$ and an element $y' \in E/F$ such that $x = a y'$, but that $E/F$ is not necessarily a divisible module (cf. Ex. 3).