Exercise 12, § 2

A $\pi$-primary module M is a direct sum of cyclic submodules if and only if there exists a pseudofree subset H in M such that, if N is the pure submodule of M generated by H, then $M(\pi) = N(\pi)$. b) The module M is a direct sum of cyclic submodules if and only if M is the union of an increasing sequence $(P_k)$ of submodules, such that the heights in M of the nonzero elements in each $P_k$ are bounded. (To see that the condition is sufficient, use a) and Ex. 11, b).) c) Deduce from b) that if M is a direct sum of cyclic submodules, then every submodule of M is a direct sum of cyclic submodules. d) Let M be a $\pi$-primary module with no elements of infinite height, having a countable system of generators $(x_i)$. Show that M is a direct sum of cyclic submodules (if

P_k is the submodule generated by x, for i ≤ k, then apply b) to the increasing sequence (P_k), using Ex. 5 applied to the decreasing sequence (P_k ∩ π^iM) (for fixed k)). e) Let M be a τ-primary module having a countable system of generators. Then M is a direct sum of indecomposable submodules if and only if the submodule N of elements of infinite height in M is pure (use d) and Ex. 7, c)).