Proposition 2

Let $A, B$ be two rings, $\rho : B \to A$ a ring homomorphism and $E$ (resp. $F$) a right (resp. left) $A$-module. Then there exists one and only one $\mathbf{Z}$-linear mapping (6)

such that, for all $x \in E$ and $y \in F$, the image under $\phi$ of the element $x \otimes y$ of $\rho_*(E) \otimes_B \rho_*(F)$ is the element $x \otimes y$ of $E \otimes_A F$; this $\mathbf{Z}$-linear mapping is surjective.

We consider the mapping $(x, y) \mapsto x \otimes y$ of $\rho_*(E) \times \rho_*(F)$ into $E \otimes_A F$; it is $\mathbf{Z}$-bilinear and, for all $\beta \in B$, by definition $(x \rho(\beta)) \otimes y = x \otimes (\rho(\beta)y)$, hence conditions (1) of no. 1 hold, whence the existence and uniqueness of $\phi$ (no. 1, Proposition 1). The latter assertion follows from the fact that the elements $x \otimes y$ generate the $\mathbf{Z}$-module $E \otimes_A F$.

The mapping (6) is called canonical.