Exercise 6, § 3

Let $A$ be a principal ideal domain with a unique maximal ideal $A \tau$, and complete with respect to the topology $\mathscr{T}$ defined in Ex. 5. a) Show that any torsion-free $A$-module $E$ of finite rank, containing no nonzero divisible submodules, is free (consider a submodule $M$ of $E$ generated by a maximal pseudofree family; if $M \neq E$ show, with the help of Ex. 4, b) and 5, b), that $E$ contains a nonzero divisible submodule). Generalise this result to the case where $E$ has countable rank (proceed by induction). b) In the free module $E = A^{(\mathbf{N})}$ (the direct sum of countably infinitely many copies of $A$), let $(a_n)_{n=1}^\infty$ be the canonical basis, and let $e_n = a_n - \pi a_n$ for each integer $n \geq 1$. Show that the submodule $M$ of $E$ generated by the $e_n$ is pure, and that the $e_n$ form a maximal pseudofree family, but that $E/M$ is a divisible module, and hence $M$ does not admit a complement in $E$.