Exercise 2, § 4

Let $A$ be the ring $\mathbf{Z}/4\mathbf{Z}$, $m$ the ideal $2\mathbf{Z}/4\mathbf{Z}$ of $A$ and $E$ the $A$-module

A/m. Show that the canonical mapping $E^* \otimes_A E \to \mathrm{Hom}_A(E, E)$ is neither injective nor surjective. The same is true of the canonical mapping

and the canonical mapping $E^* \otimes_A E^* \to (E \otimes_A E)^*$.