Exercise 1, § 4

Let $A, B$ be two rings, $E$ a left $A$-module, $F$ a right $A$-module and $G$ an $(A, B)$-bimodule; $\mathrm{Hom}_B(F, G)$ then has a left $A$-module structure and $\mathrm{Hom}_A(E, G)$ a right $B$-module structure. Show that the $\mathbf{Z}$-modules

are both canonically isomorphic to the $\mathbf{Z}$-module of $\mathbf{Z}$-bilinear mappings $f$ of $E \times F$ into $G$ such that $f(\alpha x, y) = \alpha f(x, y)$ and $f(x, y\beta) = f(x, y)\beta$ for $\alpha \in A, \beta \in B, x \in E, y \in F$.