Remark

Let $M, N, P$ be three $A$-modules, $f : M \times N \to P$ an $A$-bilinear mapping and $S^{-1}f : (S^{-1}M) \times (S^{-1}N) \to (S^{-1}P)$ the $S^{-1}A$-bilinear mapping obtained from $f$ by extending the ring of scalars to $S^{-1}A$ (Algebra, Chapter IX, § 1, no. 4, Proposition 1). Let $i : M \to S^{-1}M, j : N \to S^{-1}N$ be the canonical homomorphisms; it follows immediately that, if $Q$ is the sub-$A$-module of $P$ generated by $f(M \times N)$, $S^{-1}Q$ is the sub-$S^{-1}A$-module of $S^{-1}P$ generated by $(S^{-1}f)(i(M) \times j(N))$.