Corollary

Let $\mathfrak{J}$ be a two-sided ideal of $A$ such that $\mathfrak{J}$ is contained in the annihilator of $E$ and in the annihilator of $F$, so that $E$ (resp. $F$) has a canonical left (resp. right) $(A/\mathfrak{J})$-module structure ($§ 1$, no. 12). Then the canonical homomorphism (6)

corresponding to the canonical homomorphism $\rho : A \to A/\mathfrak{J}$ is the identity.

For all $\bar{\alpha} \in A/\mathfrak{J}$, all $x \in E$ and all $y \in F$, $x \bar{\alpha} = x \alpha$ (resp. $\bar{\alpha} y = \alpha y$) for all $\alpha$ such that $\rho(\alpha) = \bar{\alpha}$. If $C = \mathbf{Z}^{(E \times F)}$, the submodule of $C$ generated by the elements $(x \alpha, y) - (x, \alpha y)$ is then equal to the submodule generated by the elements $(x \bar{\alpha}, y) - (x, \bar{\alpha} y)$.

With the hypotheses and notation of Proposition 2, let $E'$ be a right $B$-module, $F'$ a left $B$-module and consider two semi-linear mappings $u : E' \to E$, v:F' \to F relative to the homomorphism $\rho : B \to A$; $u$ (resp. $v$) can be considered as a $B$-linear mapping $E' \to \rho_*(E)$ (resp. $F' \to \rho_*(F')$), whence a $\mathbf{Z}$-linear mapping $w : E' \otimes_B F' \to \rho_*(E) \otimes_B \rho_*(F)$ such that $w(x' \otimes y') = u(x') \otimes v(y')$ for $x' \in E',\ y' \in F'$; composing the canonical mapping (6) with this mapping, a $\mathbf{Z}$-linear mapping $w' : E' \otimes_B F' \to E \otimes_A F$ is hence obtained such that $w'(x' \otimes y') = u(x') \otimes v(y')$ for $x' \in E',\ y' \in F'$; this is the mapping which will normally be denoted by $u \otimes v$ if no confusion can arise. Clearly $(u, v) \mapsto u \otimes v$ is a $\mathbf{Z}$-bilinear mapping

Moreover, if $C$ is a third ring, $\sigma : C \to B$ a homomorphism, $E''$ a right $C$-module, $F''$ a left $C$-module, $u' : E'' \to E'$ and $v' : F'' \to F'$ semi-linear mappings relative to $\sigma$, then