Exercise 33, § 7

Let $A$ be an integral domain. An $A$-module $E$ is called divisible if for all $x \in E$ and all $\alpha \neq 0$ in $A$, there exists $y \in E$ such that $\alpha y = x$.

(a) Let $E, F$ be two $A$-modules. Show that if $E$ is divisible, so is $E \otimes_A F$. If $E$ is divisible and $F$ is torsion-free, $\operatorname{Hom}_A(E, F)$ is torsion-free. If $E$ is a torsion $A$-module and $F$ is divisible, then $E \otimes_A F = \{0\}$. If $E$ is a torsion $A$-module and $F$ is torsion-free, then $\operatorname{Hom}_A(E, F) = \{0\}$.

(b) Show that every injective $A$-module ($§ 2$, Exercise 11) is divisible and conversely that every divisible torsion-free $A$-module is injective (use criterion ($\delta$) of $§ 2$, Exercise 11).