Scholium

An element of E $\otimes_A$ F may in general be written in several ways in the form $\sum_i (x_i \otimes y_i)$, where $x_i \in E$ and $y_i \in F$; but to define a linear mapping g of the multimodule E $\otimes_A$ F into a multimodule G, there is no need to verify that, if $\sum_i (x_i \otimes y_i) = \sum_j (x'_j \otimes y'_j)$, then $\sum_i g(x_i \otimes y_i) = \sum_j g(x'_j \otimes y'_j)$; it suffices to be given $g(x \otimes y)$ for $x \in E$ and $y \in F$ and to verify that $(x, y) \mapsto g(x \otimes y)$ is $\mathbf{Z}$-bilinear and satisfies conditions (1) (no. 1) and (8).

Let E' be a ((B'_i), A, (C'_j))-multimodule, F' an (A, (B''_h); (C''_k))-multimodule and $u : E \to E', v : F \to F'$ linear mappings of multimodules; it follows immediately from the definitions (no. 2) that $u \otimes v$ is a linear mapping of the multimodule E $\otimes_A$ F into the multimodule E' $\otimes_A$ F'.

With E always denoting a right A-module, let $_sA_d$ denote the ring A considered as an (A, A)-bimodule (§ 1, no. 14, Example 1); by the above, the tensor product E $\otimes_A (_sA_d)$ has a canonical right A-module structure such that $(x \otimes \lambda)\mu = x \otimes (\lambda \mu)$ for $x \in E, \lambda \in A, \mu \in A$. The mapping $(x, \lambda) \mapsto x \lambda$ of E $\times (_sA_d)$ into E is $\mathbf{Z}$-bilinear and satisfies conditions (1) (no. 1) and (8) (where, in the latter, the $B'_i, C'_j$ and $B''_h$ are absent and the family $(C''_k)$ reduces to $A$); hence (Proposition 3), there exists an $A$-linear mapping $g$ (called canonical) of $E \otimes_A (sA_d)$ into $E$ such that $g(x \otimes \lambda) = x\lambda$ for $x \in E, \lambda \in A$.