Proposition 8

Let $M$ be a sub-A-module of $B$. The following conditions are equivalent: (a) $M$ is non-degenerate. (b) $M$ meets $S$. (c) If $j : M \to B$ is the canonical injection, the homomorphism $u = S^{-1}j : S^{-1}M \to B$ is bijective.

(a) implies (b), for if $B.M = B$, there exists $a \in A, s \in S$ and $x \in M$ such that $(a/s)x = 1$, hence $ax = s$ belongs to $S \cap M$. To see that (b) implies (c), note that $u$ is already injective ($§ 2$, no. 4, Theorem 1); moreover, if $x \in M \cap S$, the image under $u$ of $x/x \in S^{-1}M$ in $B$ is equal to 1 and $u$ is therefore surjective. Finally, (c) clearly implies (a).