Proposition 1

Let $E$ be a left $A$-module and $F$ a right $B$-module; put $E' = A' \otimes_A E$ and $F' = F \otimes_B B'$. For every bilinear mapping $\Phi$ of $E \times F$ into $G$, there exists one and only one bilinear mapping $\Phi'$ of $E' \times F'$ into $G'$ such that

whatever $a' \in A'$, $b' \in B'$, $x \in E$, $y \in F$.

The uniqueness of $\Phi'$ follows from the fact that the elements $a' \otimes x$ and $y \otimes b'$ generate $E'$ and $F'$ respectively. To prove its existence, consider the mapping

from $A' \times E \times F \times B'$ into $G$; it is obviously $\mathbf{Z}$-multilinear, and it satisfies

and $(a \in A, b \in B, a' \in A', b' \in B', x \in E, y \in F)$.

There therefore exists a $\mathbf{Z}$-bilinear mapping $\Phi'$ from $E' \times F'$ into $G'$ satisfying (13) (Chap. III, 2nd ed., App. II, No. 1, prop. 2). This relation and the definition of the module structures of $E'$ and $F'$ by (12) show that $\Phi'$ is bilinear, which completes the proof.

The hypotheses and notations being those of Proposition 1, let us now study the associated linear mappings of $\Phi$ and $\Phi'$ (No. 1, def. 2). For this purpose we shall first define a canonical homomorphism from $\mathscr{L}_A(E, G)$ into $\mathscr{L}_{A'}(E', G')$. For every $\nu \in \mathscr{L}_A(E, G)$ the mapping $(a', x) \to a'.u(\nu(x))$ from $A' \times E$ into $G'$ is $\mathbf{Z}$-bilinear, and, viewed (11), maps $(a'h(a), x)$ and $(a', ax)$ $(a \in A)$ onto the same element of $G'$; it therefore defines (Chap. III, 2nd ed.,

App. II, Nos. 1 and 10) a mapping $k(\nu)$ from $E' = A' \otimes_A E$ into $G'$ such that $k(\nu)(a' \otimes x) = a'.u(\nu(x))$, and which, viewed (12), is $A'$-linear. Moreover one immediately deduces from (12) that the mapping $\nu \to k(\nu)$ from $\mathscr{L}_A(E, G)$ into $\mathscr{L}_{A'}(E', G')$ satisfies $k(\nu b) = k(\nu)h'(b)$ for every $b \in B$. Let $i$ denote the canonical mapping $y \to y \otimes 1$ from $F$ into $F'$. Then the diagram

(where $d_\Phi$ and $d_{\Phi'}$ denote the right associated linear mappings to $\Phi$ and $\Phi'$) is commutative. Indeed, for $x \in E, y \in F$ and $a' \in A'$, we have $d_{\Phi'}(i(y))(a' \otimes x) = \Phi'(a' \otimes x, y \otimes 1) = a'.u(\Phi(x, y)) = a'.u(d_\Phi(y)(x))$, that is, $d_{\Phi'}(i(y))(a' \otimes x) = k(d_\Phi(y))(a' \otimes x)$. There is an analogous commutation relation for the left associated linear mappings $s_\Phi$ and $s_{\Phi'}$ to $\Phi$ and $\Phi'$.