Theorem 1

The ring $S^{-1}A$ is a flat $A$-module (Chapter I, § 2, no. 1, Definition 2).

If $u : M' \to M$ is an injective homomorphism of $A$-modules, it is necessary to establish that $S^{-1}u : S^{-1}M' \to S^{-1}M$ is injective. Now, if $m'/s$ ($m' \in M'$, $s \in S$) is such that $u(m')/s = 0$, this implies the existence of an $s' \in S$ such that $s'u(m') = 0$ (no. 2, Proposition 4) or $u(s'm') = 0$; as $u$ is injective, it follows that $s'm' = 0$, whence $m'/s = 0$.

The fact that $S^{-1}A$ is a flat $A$-module allows us to apply to it the results of Chapter I, § 2. In particular: (1) If $M$ is an $A$-module and $N$ a submodule of $M$, $S^{-1}N$ is canonically identified with a submodule of $S^{-1}M$ generated by $i_M^S(N)$ (Chapter I. § 2, no. 3, Remark 2); with this identification, $S^{-1}(M/N)$ is identified with $(S^{-1}M)/(S^{-1}N)$ and, if $P$ is a second submodule of $M$, then

(Chapter I, § 2, no. 6, Proposition 2). (2) If $M$ is a finitely generated $A$-module, then

(Chapter I, § 2, no. 10, Corollary 2 to Proposition 12).