Remark

(1) The above remarks concerning the case of modules over a commutative ring generalize as in no. 5, Remark 2 when there is a tensor product $E_1 \otimes_{A_1} E_2 \otimes \cdots \otimes_{A_{n-1}} E_n$ where the rings $A_i$ are not necessarily commutative and where, for each $i$, there is a homomorphism $\rho_i : C \to A$ of the same commutative ring $C$ such that: (1) $\rho_i(C)$ is contained in the centre of $A_i$; (2) for $2 \leq i \leq n - 1$, the $C$-module structures on $E_i$ obtained using the homomorphisms $\rho_{i-1}$ and $\rho_i$ coincide. Then a $C$-*module* structure is obtained on $E_1 \otimes_{A_1} E_2 \otimes \cdots \otimes_{A_{n-1}} E_n$ and canonical mappings analogous to (13) (no. 5), which will be left to the reader to describe.

(2) Let $A, B$ be two rings, $E$ a right $A$-module, $E'$ a left $A$-module, $F$ a right $B$-module and $F'$ a left $B$-module. The $\mathbf{Z}\text{-bilinear}$ mappings of $(E \otimes_A E') \times (F \otimes_B F')$ into a $\mathbf{Z}\text{-module}$ $G$ are then in one-to-one correspondence with the $\mathbf{Z}\text{-multilinear}$ mappings $f$ of $E \times E' \times F \times F'$ into $G$ satisfying the conditions

for $\lambda \in A, \mu \in B, x \in E, x' \in E', y \in F, y' \in F'$. The general constructions given in this no. reduce the proof of this to defining a canonical $\mathbf{Z}\text{-module}$ isomorphism between $(E \otimes_A E') \otimes_{\mathbf{Z}} (F \otimes_B F')$ and $E \otimes_A E' \otimes_{\mathbf{Z}} F \otimes_B F'$, which follows from the associativity property of tensor products of the form (33).