Exercise 10, § 3

Let M be a nonzero Z-submodule of the ring of p-adic integers $\mathbf{Z}_p$ such that M is a pure Z-submodule of the Z-module $\mathbf{Z}_p$ (VII, p. 55, Ex. 7). Show that $M + p \mathbf{Z}_p = \mathbf{Z}_p$; deduce that M is indecomposable (observe that $\mathbf{Z}_p/p \mathbf{Z}_p$ is isomorphic to $\mathbf{Z}/p \mathbf{Z}$). Give an example of a decomposable Z-submodule of $\mathbf{Z}_p$ (observe that $\mathbf{Z}_p$ has the cardinality of the continuum).