Exercise 4, § 2

A family $(x_i)$ of nonzero elements of a module M over a principal ideal domain A is said to be pseudofree if every relation of the form $\sum \alpha_i x_i = \beta y$ ($y \in M,\ a_i \in A,\ \beta \in A$) implies the existence of a family of elements $\alpha'_i \in A$ such that $\alpha_i x_i = \beta \alpha'_i x_i$ for each index $i$. The relation $i \neq k$ then implies $x_i \neq x_k$; the set of elements of a pseudofree family is called a pseudofree subset of M. a) Let $\pi$ be an irreducible element of A. Show that, in an A-module with annihilator $A\tau^n \ (n \geq 1)$, any element with annihilator $A\tau^n$ is pseudofree (use Bezout's identity). b) Let $(x_i)$ be a pseudofree family in an A-module M, and let N be the submodule of M generated by this family. Show that for every element $\dot{x}$ of $M/N$ there exists an element $x \in \dot{x}$ whose annihilator is equal to the annihilator of $\dot{x}$ in $M/N$. If also $\dot{x}$ is pseudofree in $M/N$, show that the family consisting of the $x_i$ and x is pseudofree in M. c) Deduce from a) and b) that if the A-module M has nonzero annihilator, then it is a direct sum of cyclic modules (reduce to the case of a $\tau$-primary module, and apply Zorn's Lemma to pseudofree families of elements of M) (cf. VII, p. 19, Th. 2).