Proposition 1

For every left $A$-module $E$, the mapping $\phi : x \mapsto 1 \otimes x$ of $E$ into the $A$-module $\rho_*(\rho^*(E))$ is $A$-linear and the set $\phi(E)$ generates the $B$-module $\rho^*(E)$. Further, for every left $B$-module $F$ and every $A$-linear mapping $f$ of $E$ into the $A$-module $\rho_*(F)$, there exists one and only one $B$-linear mapping $\bar{f}$ of $\rho^*(E)$ into $F$ such that $\bar{f}(1 \otimes x) = f(x)$ for all $x \in E$.

$B$ can be considered as a $(B, A)$-bimodule by means of $\rho$; then there is a canonical $\mathbf{Z}$-module isomorphism

(1)

as has been seen in $§ 4$, no. 1, Proposition 1. But the left $A$-module $\operatorname{Hom}_B(B_s, F)$ is canonically identified with $\rho^*(F)$: for, by definition ($§ 1$, no. 14), there corresponds to an element $y \in F$ the homomorphism $\theta(y) : B_s \to F$ such that $(\theta(y))(1) = y$; for all $\lambda \in A$, there thus corresponds to $\rho(\lambda)y \in F$ the homomorphism $\mu \mapsto \mu \rho(\lambda)y$ of $B_s$ into $F$, which is just $\lambda \theta(y)$ for the left $A$-module structure on $\operatorname{Hom}_B(B_s, F)$ ($§ 1$, no. 14). Using this identification, we obtain therefore a canonical $\mathbf{Z}$-module isomorphism, the inverse of (1)

(2)

and it follows immediately from the definitions that if $\delta(f) = \bar{f}$ then $\bar{f}(1 \otimes x) = f(x)$ for all $x \in E$. In particular, the mapping $\phi_E : x \mapsto 1 \otimes x$ is just (3)

Proposition 1 is therefore proved. The mapping $\phi_E : E \to \rho_*(\rho^*(E))$ is called canonical.