Exercise 3, § 2

Let $S = \mathbf{Z}^*$ (the complement of $\{0\}$ in $\mathbf{Z}$), $p$ an integer $> 1$ and $M$ the $\mathbf{Z}$-module the direct sum of the modules $\mathbf{Z}/p^n\mathbf{Z}$ for $n \in \mathbf{N}$. Show that $S^{-1}M = 0$ although $M$ is a faithful $\mathbf{Z}$-module and that the canonical mapping

is not injective.