Proposition 1

(a) Let E be a (B, A)-bimodule, F a left A-module and G a left B-module. For every mapping $g \in \mathrm{Hom}_B(E \otimes_A F, G)$, let $g'$ be the mapping of F into $\mathrm{Hom}_B(E, G)$ defined by $(g'(y))(x) = g(x \otimes y)$ for $x \in E, y \in F$. The mapping $g \mapsto g'$ is an isomorphism

(5)

(b) Let E be a right A-module, F an (A, C)-bimodule and G a right C-module. For every mapping $h \in \mathrm{Hom}_C(E \otimes_A F, G)$, let $h'$ be the mapping of E into $\mathrm{Hom}_C(F, G)$ defined by $(h'(x))(y) = h(x \otimes y)$ for $x \in E, y \in F$. The mapping $h \mapsto h'$ is an isomorphism

(6)

In particular B and C may be taken to be a subring $\Gamma$ of the centre of the ring A; then for every $\Gamma$-module G, the three $\Gamma$-modules

are canonically isomorphic to the $\Gamma$-module of $\Gamma$-*bilinear* mappings of $E \times F$ into G which satisfy (1). More particularly: