Proposition 3

Let $A$ be a ring, $S$ a subset of $A$, $M$ an $A$-module, $M'$ the $A$-module $M \otimes_A A[S^{-1}]$ and $f$ the canonical $A$-homomorphism $x \mapsto x \otimes 1$ of $M$ to $M'$. Then: (1) For all $s \in S$, the homothety $z \mapsto sz$ of $M'$ is bijective. (2) For every $A$-module $N$ such that, for all $s \in S$, the homothety $y \mapsto sy$ of $N$ is bijective, and every homomorphism $u$ of $M$ to $N$, there exists a unique homomorphism $u'$ of $M'$ to $N$ such that $u = u' \circ f$.

In other words, $(M', f)$ is a solution of the universal mapping problem (Set Theory, Chapter IV, § 3, no. 1) with the following conditions: the species of structure $\Sigma$ is that of an $A$-module in which the homotheties induced by the elements of $S$ are bijective, the morphisms are $A$-module homomorphisms and the $a$-mappings are also $A$-module homomorphisms.

For every $A$-module $N$ and all $a \in A$, denote by $h_a$ the homothety $y \mapsto ay$ in $N$; $a \mapsto h_a$ is then a ring homomorphism from $A$ to $\operatorname{End}_A(N)$. To say that $h_a$ is bijective means that $h_a$ is an invertible element of $\operatorname{End}_A(N)$. Suppose that, for all $s \in S$, $h_s$ is invertible in $\operatorname{End}_A(N)$; the elements $h_a$, where $a \in A$, and the inverses of the elements $h_s$ where $s \in S$, then generate in $\operatorname{End}_A(N)$ a commutative subring $B$ and the homomorphism $a \mapsto h_a$ from $A$ to $B$ is such that the images of the elements of $S$ are invertible. Then it follows (no. 1, Proposition 1) that there exists a unique homomorphism $h'$ of $A[S^{-1}]$ to $B$ such that

we know (Algebra, Chapter II, , no. 14) that such a homomorphism defines on N an $A[S^{-1}]$-module structure such that $(a/s).y = h_s^{-1}(a.y)$; the A-module structure derived from this $A[S^{-1}]$-module structure by means of the homomorphism $i_A^S$ is precisely the structure given initially

Conversely, if N is an $A[S^{-1}]$-module and it is considered as an A-module by means of $i_A^S$, the homotheties $y \mapsto sy$, for $s \in S$, are bijective, for $y \mapsto (1/s)y$ is the inverse mapping of $y \mapsto sy$; and the $A[S^{-1}]$-module structure on N derived from its A-module structure by the process described above is the $A[S^{-1}]$-module structure given initially. Thus there is a canonical one-to-one correspondence between $A[S^{-1}]$-modules and A-modules in which the homotheties induced by the elements of S are bijective; moreover, if N, N' are two A-modules with this property every A-module homomorphism $u : N \to N'$ is also a homomorphism of the $A[S^{-1}]$-module structures of N and N', as, for ally $\in N$ and all $s \in S$, we may write $u(y) = u(s.((1/s)y)) = s.u((1/s)y)$, whence $u((1/s)y) = (1/s)u(y)$; the converse is obvious.

This being so, the statement of Proposition 3 is just the characterization of the module obtained from M by extending the scalars to $A[S^{-1}]$, taking account of the above interpretation (Algebra, Chapter II, § 5, no. 1, Remark 1).