Remark

(1) Let A, B be two commutative rings, $\rho : B \to A$ a ring homomorphism and E and F two A-modules; then the canonical mapping (6) of no. 3 is a B-linear mapping

(2) What has been said in this no. can be generalized to the following case: let E be a right A-module, F a left A-module, C a commutative ring and $\rho : C \to A$ a homomorphism of C into A such that $\rho(C)$ is contained in the centre of A (cf. III, § 1, no. 3). We may then consider the C-modules $\rho_*(E)$ and $\rho_*(F)$ and the hypothesis on $\rho$ implies that these C-module structures are compatible respectively with the A-module structures on E and F (§ 1, no. 14). The tensor product $E \otimes_A F$ is thus (by virtue of no. 4) given two C-module structures such that $\gamma(x \otimes y) = (x \rho(\gamma)) \otimes y$ and

respectively for $\gamma \in C$, $x \in E$, $y \in F$ and Definition 1 (no. 1) shows also that these two structures are identical. If E' (resp. F') is a right (resp. left) A-module and $u : E \to E'$, $v : F \to F'$ are two A-linear mappings, then $u \otimes v : E \otimes_A F \to E' \otimes_A F'$ is C-linear for the C-module structures just defined; the mapping $(u, v) \mapsto u \otimes v$:

is C-bilinear (for the C-module structures on $\mathrm{Hom}_A(E, E')$ and $\mathrm{Hom}_A(F, F')$) defined in § 1, no. 14, Remark 1), whence we also derive a C-linear mapping, called canonical

(13) $\operatorname{Hom}_A(E, E') \otimes_C \operatorname{Hom}_A(F, F') \to \operatorname{Hom}_C(E \otimes_A F, E' \otimes_A F')$.

(3) Let A be an integral domain and K its field of fractions. If E and F are two vector K-spaces, the canonical mapping

(no. 3 and § 1, no. 13) is bijective. It suffices (no. 4) to prove that if $f$ is an A-bilinear mapping of $E \times F$ into a vector K-space G, $f$ is also K-bilinear. Now, for all $\alpha \neq 0$ in A.

whence since G is a vector K-space.