Exercise 6, § 4

Let $A, B$ be two rings, $E$ a left $A$-module, $F$ an $(A, B)$-bimodule and $G$ a right $B$-module. (a) Show that there exists one and only one $\mathbf{Z}$-linear mapping

such that for $x \in E$ and $u \in \mathrm{Hom}_B(F, G)$, $\sigma(u \otimes x)$ is the mapping $v \mapsto u(v(x))$ of $\mathrm{Hom}_A(E, F)$ into $G$. (b) If $E$ is a finitely generated projective $A$-module, show that $\sigma$ is bijective. (c) Suppose that $G$ is an injective $B$-module (§ 2, Exercise 11) and that $E$ is the cokernel of an $A$-linear mapping $A_s^m \to A_s^n$. Show that $\sigma$ is bijective (start from the exact sequence $A_s^m \to A_s^n \to E \to 0$ and use (b), the definition of injective modules and Theorem 1 of § 2, no. 1, and Proposition 5 of § 3, no. 6).