Proposition 8

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

such that $\phi((x \otimes y) \otimes z) = x \otimes (y \otimes z)$ for $x \in E,\ y \in F,\ z \in G;$ moreover this $\mathbf{Z}$-linear mapping is bijective ("associativity" of the tensor product).

The right B-module structure on E ⊗_A F and left A-module structure on F ⊗_B G have been defined in no. 4. The uniqueness of $\phi$ is obvious since the elements $(x \otimes y) \otimes z$ generate the $\mathbf{Z}$-module $(E \otimes_A F) \otimes_B G$. To show the existence of $\phi$, we note that, for all $z \in G,\ h_z : y \mapsto y \otimes z$ is an A-linear mapping of the left A-module F into the left A-module F ⊗_B G. We write $g_z = 1_E \otimes h_z$, which is therefore a $\mathbf{Z}$-linear mapping of E ⊗_A F into E ⊗_A (F ⊗_B G) and consider the mapping $(t, z) \mapsto g_z(t)$ from $(E \otimes_A F \times G$ into $E \otimes_A (F \otimes_B G)$; as $h_{z + z'} = h_z + h_{z'}$ for $z \in G,\ z' \in G$, it is immediate that the above mapping is $\mathbf{Z}$-bilinear. Further, we show that for all $\mu \in B,\ g_{\mu z}(t) = g_z(t \mu)$; it is obviously sufficient to do this for $t = x \otimes y$ where $x \in E$ and $y \in F$; now and

Proposition 1 (no. 1) then proves the existence of a $\mathbf{Z}$-linear mapping

such that $\phi(t \otimes z) = g_z(t)$, hence $\phi((x \otimes y) \otimes z) = x \otimes (y \otimes z)$. Similarly a $\mathbf{Z}$-linear mapping

is defined such that $\psi(x \otimes (y \otimes z)) = (x \otimes y) \otimes z$ and clearly $\psi \circ \phi$ and $\phi \circ \psi$ are the identity mappings of $(E \otimes_A F) \otimes_B G$ and $E \otimes_A (F \otimes_B G)$ respectively, since they reduce to the identity mappings on generating systems of these $\mathbf{Z}$-modules.

It is immediate that, if $E$ is a $((C'_i); A, (D'_j))$-multimodule, $F$ an $(A, (C''_h); B, (D''_k))$-multimodule and $G$ a $(B, (C'''_l); (D'''_m))$-multimodule, the canonical isomorphism defined in Proposition 8 is a $((C'_i), (C''_h), (C'''_l); (D'_j), (D''_k), (D'''_m))$-multimodule isomorphism. In particular, if $C$ is a commutative ring and $E, F, G$ three $C$-modules, there is a canonical $C$-module isomorphism

We shall see below that, under certain conditions, the definition of tensor product can be generalized to a family of multimodules, which will in particular give us under the hypotheses of Proposition 8 a $\mathbf{Z}$-module $E \otimes_A F \otimes_B G$, which is canonically isomorphic to each of the $\mathbf{Z}$-modules $(E \otimes_A F) \otimes_B G$ and $E \otimes_A (F \otimes_B G)$ and with which the latter will be identified.