Exercise 9, § 3

Let A be a principal ideal domain and let M be the dual of the A-module $A^N$. Let $f : M \to A^N$ be the transpose map of the canonical injection $A^{(N)} \to A''$ (where $A^N$ is identified with the dual of $A^{(N)}$). a) Let $\pi$ be an irreducible element of A and let $\varphi : A^N/A^{(N)} \to A$ be a linear form; show that if $u \in A^N/A^{(N)}$ is the class of an element $(u_n) \in A''$ such that $u_n \in (\pi^n)$ for all $n$, then $\varphi(u) = 0$. If A contains two non-associate irreducible elements, deduce that f is injective. (Using the above and Bezout's identity, show that the dual of $A^N/A^{(N)}$ is zero.) b) Give A the topology defined in Ex. 5 of VII, p. 60. Show that, if f(M) is not contained in $A^{(N)}$, then A is complete. (Reduce the problem to showing that, if $\varphi : A^N \to A$ is a linear form such that $\varphi(e_n) \neq 0$ for every element $e_n$ of the canonical basis of $A^{(N)}$, then $\operatorname{Im}(\varphi)$ contains the sum of any series of elements of A which tends to 0 sufficiently quickly.) c) If A contains two non-associate irreducible elements, then the canonical map from $A^{(N)}$ (resp. A') into its double dual is bijective (use a) and b)).