Exercise 4, § 2

Let $S = \mathbf{Z}^*$ and $M$ be a free $\mathbf{Z}$-module with infinite basis. Show that the canonical mapping $S^{-1}\operatorname{End}_\mathbf{Z}(M) \to \operatorname{End}_{S^{-1}\mathbf{Z}}(S^{-1}M)$ is not surjective. Deduce from this and Exercise 3 an example of a $\mathbf{Z}$-module $M$ such that the canonical mapping $S^{-1}\operatorname{End}_\mathbf{Z}(M) \not\to \operatorname{End}_{S^{-1}\mathbf{Z}}(S^{-1}M)$ is neither injective nor surjective.