Exercise 5, § 4

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

which maps the tensor product $x \otimes y$, where $x \in E$ and $y \in F$, to the mapping $v_{x, y} : u \mapsto u(x) \otimes y$ of $\mathrm{Hom}_A(E, G)$ into $G \otimes_A F$. When $B = A$ and $G = {}_sA_d$, the homomorphism reduces to the canonical homomorphism (15) (no. 2). (b) Show that if $F$ is a projective $A$-module and $E$ and $G$ are such that for $x \neq 0$ in $E$ there exists $u \in \mathrm{Hom}_A(E, G)$ such that $u(x) \neq 0$, then the homomorphism $\eta$ is injective.