Exercise 29, § 7

Let $A$ be an integral domain, $K$ its field of fractions, $E$ an $A$-module and $T(E)$ its torsion module.

(a) Show that every linear form on $E$ is zero on $T(E)$, so that the duals $E^*$ and $(E/T(E))^*$ are canonically isomorphic.

(b) Show that the canonical mapping $(E^*)_{(K)} \to (E_{(K)})^*$ is injective and that it is bijective when $E$ is finitely generated.

(c) Give an example of a torsion-free $A$-module $E$ such that $E^* = \{0\}$ and $E_{(K)} \neq \{0\}$.