Exercise 6, § 2

Let M be a torsion module over a principal ideal domain A, which is not finitely generated, but all of whose proper submodules are finitely generated. Show that there exists an irreducible element $\tau_1$ of A such that M is isomorphic to the $\tau$-primary component $U_{\pi}$ of the module $U = K/A$ (Ex. 3, c)). (First of all show that M cannot have more than one nonzero $\pi$-primary component, and hence is a $\tau$-primary module for some irreducible element $\tau \in A$. Then show that $M = \pi M$ by observing that $M/\pi M$ could not be finitely generated if it were nonzero; but $M/\pi M$ is a vector space over the field $A/\pi A$, and would contain a proper subspace which was not finitely generated.)