Exercise 5, § 3

Let $A$ be a principal ideal domain having a unique maximal ideal $A \pi$ (cf. VII, p. 48, Ex. 4). Let $E$ be a torsion-free $A$-module, and $M$ a free submodule of $E$, such that $E/M$ is torsion-free, divisible and of rank 1. Let $(e_\alpha)$ be a basis of $M$, let $u$ be a nonzero element of $E/M$, and $u$ an element of the coset $u$; for each integer $n > 0$, let $u_n$ be an element of $E$ such that $u \equiv \pi^n u_n \pmod{M}$. Put $u = \pi^n u_n + \sum \lambda_{n\alpha} e_\alpha$.

a) Show that $M$ is generated by $M$ and the $u_n$, and that $A_\alpha - A_{\alpha-1,\alpha} \in A n^n$ for all $\alpha$ and for every integer $n > 0$ (cf. Ex. 4, a)). b) Suppose in addition that $A$ is complete with respect to the topology $\mathscr{T}$ defined by taking the ideals $A n^n$ as a basis of open neighbourhoods of $0$ in $A$ (Gen. Top., III, p. 5); put $A_\alpha = \lim_{n \to \infty} A_{\alpha-1,\alpha}$. Show that, if $\mu$ and $\mu_\alpha$ are such that $\mu u - \sum \mu_\alpha e_\alpha$ belongs to all the submodules $\pi^n E$, then necessarily $\mu_\alpha = \lambda_\alpha \mu$ for all $\alpha$.