Proposition 3

Let G be a ((B'_i), (B''_h); (C'_j), (C''_k))-multimodule.

(a) Let g be a linear mapping of the multimodule E $\otimes_A$ F into G. The mapping $f : (x, y) \mapsto g(x \otimes y)$ of E $\times$ F into G is $\mathbf{Z}$-bilinear and satisfies relations (1) of no. 1 and the conditions

for all $x \in E, y \in F, \mu'_i \in B'_i, v'_j \in C'_j, \mu''_h \in B''_h, v''_k \in C''_k, i, j, h, k$ arbitrary.

(b) Conversely, let f be a $\mathbf{Z}$-bilinear mapping of E $\times$ F into G satisfying conditions (1) (no. 1) and (8). Then there exists one and only one linear mapping g of the multimodule E $\otimes_A$ F into the multimodule G such that $f(x, y) = g(x \otimes y)$ for $x \in E, y \in F$.

Assertion (a) follows immediately from the definition of the multimodule structure on E $\otimes_A$ F, since for example $(x \otimes y)v'_j = (xv'_j) \otimes y$. To prove (b), we note first that Proposition 1 of no. 1 gives the existence and uniqueness of a $\mathbf{Z}$-linear mapping g such that $g(x \otimes y) = f(x, y)$ for $x \in E, y \in F$; all that is needed is to verify that g is linear for the multimodule structures. As the elements $x \otimes y$ generate the $\mathbf{Z}$-module E $\otimes_A$ F, it suffices to verify the relations $g(\mu'(x \otimes y)) = \mu'_i g(x \otimes y)$ and their analogues; but this follows immediately from the formula $g(x \otimes y) = f(x, y)$ and relations (8).