Exercise 9, § 4

Let A be a ring, C its centre, E a right A-module, F a left A-module and E*, F* the respective duals of E and F. An additive mapping $f$ of $E^* \otimes_C F^*$ into A is called doubly linear if $f(aw) = af(w)$ and $f(wa) = f(w)a$ for all $w \in E^* \otimes_C F^*$ and all $a \in A$. Show that there exists one and only one C-linear mapping $\phi$ of $E \otimes_A F$ into the C-module L of doubly linear mappings of $E^* \otimes_C F^*$ into A such that

for all $x \in E, y \in F, x^* \in E^*, y^* \in F^*$. If E and F are finitely generated projective modules, $\phi$ is bijective.