Exercise 10, § 2

u) Let $M$ be a $n$-primary module with no nonzero elements of infinite height and let $N$ be a pure submodule of $M$ generated by a maximal pseudofree family of elements of $M$ (Ex. 4 and 7, b)). Show that $M/N$ is a divisible module (argue by contradiction, applying Ex. 8, c), Ex. 7, c), Ex. 7, b), and finally Ex. 4, c)).

b) Consider the metrisable topology $\mathscr{T}$ on M defined by taking as a basis of neighbourhoods of 0 the submodules $\pi^n M$ for all integers $n \geq 0$ (Gen. Top., III, p. 5). Then the quotient $M/P$ of M by a submodule P is divisible if and only if P is everywhere dense in M under the topology $\mathscr{T}$.

c) Let $P_1, P_2$ be two submodules of M which are pure, everywhere dense in M under the topology $\mathscr{T}$, and direct sums of cyclic submodules. Show that $P_1$ and $P_2$ are isomorphic (notice that $P_1 / \pi^n P_1$ is isomorphic to $M / \pi^n M$ for all n).

d) Let M be the completion of M with respect to the topology $\mathscr{T}$ (Gen. Top., III, p. 24). Show that $x \mapsto \alpha x$ is a strict morphism from M onto $\alpha \hat{M}$ for all $a \in A$, that $\pi^n \hat{M}$ is an open and closed submodule, and the closure of $\pi^n M$, for all $n$, and that the quotient modules $\hat{M} / \pi^n \hat{M}$, $M / \pi^n M$ are isomorphic.

e) Let M be the torsion submodule of $\dot{M}$: show that M is a =-primary module, that $M = \overline{M}$, and that M is a pure submodule of M.

f) Suppose the $\pi$-primary module M is the direct sum of a family $(E_i)$ of cyclic submodules. Then $\mathscr{T}$ is discrete if and only if the annihilator of M is 0. In this case, let E be the product module $\prod E_a$ and let F be the submodule of E consisting of elements all but countably many of whose coordinates are zero; show that M can then be identified (as a non-topological module) with the submodule of F consisting of $(x_i)$ such that, for each integer $n$, there are only finitely many indices $\alpha$ for which $x_\alpha \notin \pi^n E_\alpha$. Then show that the four modules M, $\hat{M}$, M and F are distinct. Deduce that M is not a direct sum of cyclic modules (use e)), and that it is not a direct sum of indecomposable modules (notice that $\overline{M}$ contains no elements of infinite height, and apply Ex. 8, c)).