Example

The injective $\mathbf{Z}$-module $\mathbf{Q}/\mathbf{Z}$ (X, p. 18, example 2) is a cogenerator. Indeed, every $\mathbf{Z}$-module simple module is isomorphic to a module $\mathbf{Z}/p\mathbf{Z}$, $p\ne 0$, and $\operatorname{Hom}_{\mathbf{Z}}(\mathbf{Z}/p\mathbf{Z},\mathbf{Q}/\mathbf{Z})$ is nonzero (it contains for example the element of $\mathbf{Z}/p\mathbf{Z}$ obtained by passing to quotients from the homomorphism $x\mapsto x/p$ of $\mathbf{Z}$ into $\mathbf{Q}$).