Exercise 5, § 5

Take $A = \mathbf{Z}$ and $3 = p\mathbf{Z}$, where $p$ is prime. (a) Show that, if $q$ is a prime number distinct from $p$, the $\mathbf{Z}$-module $\mathbf{Z}/q\mathbf{Z}$ satisfies property (ii) of Theorem 1, but not property (i). (b) Show that the $\mathbf{Z}$-module $\mathbf{Q}/\mathbf{Z}$ satisfies property (iv) of Theorem 1 but not property (ii).