Exercise 8, § 3

Let A be a principal ideal domain which is not a field. a) Show that the product A-module A'' admits no countable system of generators. (If A is countable, consider Card(A^N); otherwise notice that, if for all x \in A we let v_x denote the element $(x^n)_{n \geq 0}$ of A'', then the $v_x$ form a free system). b) Let $\pi$ be an irreducible element of A. Let S denote the submodule of A'' consisting of those sequences $z = (z_n)_{n \in \mathbf{N}}$ for which there exists a sequence $(k(n))_{n \in \mathbf{N}}$ of integers tending to $+\infty$ (depending on z), such that $z_n \in A \pi^{k(n)}$ for all n. Show that the $(A/\pi)$-vector space $S/\pi S$ has a countable basis. c) Deduce from a) and b) that $A^N$ is not a free A-module. (Otherwise, S would be a free A-module with a countable basis; but $(x,$ $(x_n \pi^n)$ is an injective A-linear map from $A^N$ into S.)