Exercise 1, § 5

Define an endomorphism $u$ of the $\mathbf{Z}$-module $M = \mathbf{Z}^{(\mathbf{N})}$ such that, for the prime ideal $p = (0)$ of $\mathbf{Z}$, $u_p$ is an automorphism of $M$, but that there exists no $f \neq 0$ in $\mathbf{Z}$ such that $u_f$ is a surjective endomorphism of $M_f$.