Exercise 11, § 2

Let M be a $\tau$-primary module and let N be a submodule of M whose nonzero elements all have height $\leq h$ in M. a) Let P be a pure submodule of M such that $P(\pi) \subset N(\pi)$ and $P(\tau) \neq N(\pi)$. Let a be an element of $N(\pi)$ not belonging to $P(\pi)$, and let $x_0$ be an element of $a + P(\pi)$ whose height k in M is as large as possible; let $x_0 = \pi^k y_0$. Show that the sum $P + Ay$, is direct, and is a pure submodule of M. b) Let B be a pseudofree subset of M such that, if P is the pure submodule generated by B, then $P(\pi) \subset N(\pi)$. Show that there exists a pseudofree subset $C \to B$ of M such that, if Q is the pure submodule of M generated by C, then $Q(\pi) = N(\pi)$ (apply Zorn's Lemma, using a)).