Exercise 6, § 5

(a) Let A be a valuation ring, M an A-module and M' a submodule of M. Show that for M' to be a pure submodule of M (Chapter I, § 2, Exercise 24), it is necessary and sufficient that, for all $a \in A$, $M' \cap (\alpha M) = \alpha M'$.

(b) Let M' be a pure submodule of M such that, if M' is given the topology for which the $\alpha M'$ (where $a \in A, a \neq 0$) form a fundamental system of neighbourhoods of 0, M' is linearly compact. Show that for every element $x \in M$ there exists $y' \in M'$ such that $x' = x + y'$ has the following property: for all $a \in A$ such that $x + M' \in \alpha(M/M')$, $x' \in \alpha M$. (For each of the $a \in A$ such that $x + M' \in \alpha(M/M')$, consider the subset S, of M' consisting of the $y'_\alpha$ such that $x + y'_\alpha \in \alpha M$.)

(c) Suppose that A is a linearly compact valuation ring; let K be the field of fractions of A. Show that, if M is a torsion-free A-module such that $M_{(K)}$ admits a countable basis, M is the direct sum of a countable family of A-modules of rank 1. (Consider M as the union of an increasing sequence $(M'_i)$ of pure submodules such that $M'_i$ is of rank i and for each i apply (b) to $M'_i$ and its submodule $M'_{i-1}$.

(d) With the same hypotheses on A and M as in (c), let N be a pure submodule of M of finite rank. Show that N is a direct factor of M (observe that every A-module which is the direct sum of a finite number of A-modules of rank 1 is linearly compact and use (b)).

(e) With the same hypotheses on M and M' as in (b), show that for all $x \in M$ there exists $y' \in M'$ such that the annihilator of $x' = x + y'$ is equal to the annihilator of the element $x + M' \in M/M'$. (Let a be the annihilator of $x + M'$; for all $a \in a$, let T, be the subset of M' consisting of they' such that $a(x + y') = 0$; show that the intersection of the T, is non-empty.)

(f) Suppose that A is a valuation ring such that for every ideal $a \neq 0$ the A-module A/a is linearly compact (with the discrete topology). Show that every finitely generated torsion A-module M is the direct sum of a finite number of monogenous A-modules. (If $(z_i)_{1 \leq i \leq n}$ is a system of generators of M, argue by induction on n, considering one of the $z_i$ whose annihilator is the least and noting that the submodule of M which it generates is pure.)