Proposition 1

(a) Let $g$ be a $\mathbf{Z}$-linear mapping of $E \otimes_A F$ into a $\mathbf{Z}$-module $G$. The mapping $(x, y) \mapsto f(x, y) = g(x \otimes y)$ of $E \times F$ into $G$ is $\mathbf{Z}$-bilinear and satisfies conditions (1).

(b) Conversely, let $f$ be a $\mathbf{Z}$-bilinear mapping of $E \times F$ into a $\mathbf{Z}$-module $G$ satisfying conditions (1). Then there exists one and only one $\mathbf{Z}$-linear mapping $g$ of $E \otimes_A F$ into $G$ such that $f(x, y) = g(x \otimes y)$ for $x \in E, y \in F$.

If $\phi$ denotes the canonical mapping of $E \times F$ into $E \otimes_A F$, then $f = g \circ \phi$; whence (a). To show (b), we note that, in the notation of Definition 1, $f$ extends to a $\mathbf{Z}$-linear mapping $\bar{f}$ of $C$ into $G$ (§ 1, no. 11, Proposition 17). By virtue of relations (1), $\bar{f}$ is zero for all the elements of $C$ of one of the types (2) and hence on $D$. There therefore exists a $\mathbf{Z}$-linear mapping $g$ of $C/D = E \otimes_A F$ into $G$ such that $\bar{f} = g \circ \psi$, where $\psi : C \to C/D$ is the canonical homomorphism (§ 1, no. 8, Remark). The uniqueness of $g$ is immediate since $E \otimes_A F$ is generated, as a $\mathbf{Z}$-module, by the elements of the form $x \otimes y$.

Proposition 1 defines a canonical isomorphism of the $\mathbf{Z}$-module of $\mathbf{Z}$-bilinear mappings $f$ of $E \times F$ into $G$, satisfying conditions (1), onto the $\mathbf{Z}$-module $\mathrm{Hom}_{\mathbf{Z}}(E \otimes_A F, G)$.

When $A = \mathbf{Z}$, conditions (1) are automatically satisfied for every $\mathbf{Z}$-bilinear mapping $f$ and the submodule $D$ of $C$ is already generated by the elements of the first two types in (2).

If now we return to the general case and $E'$ and $F'$ denote the underlying $\mathbf{Z}$-modules of $E$ and $F$ respectively, the above remark and Definition 1 show immediately that the $\mathbf{Z}$-module $E \otimes_A F$ can be canonically identified with the quotient of the $\mathbf{Z}$-module $E' \otimes_{\mathbf{Z}} F'$ by the sub-$\mathbf{Z}$-module generated by the elements of the form $(x \lambda) \otimes y - x \otimes (\lambda y)$, where $x$ runs through $E$, $y$ runs through $F$ and $\lambda$ runs through $A$.