Proposition 9

Let $M, N$ be two sub-A-modules of $B$. If $M$ is non-degenerate, the canonical homomorphism from $N : M$ to $\operatorname{Hom}_A(M, N)$ is bijective.

Let $s \in S \cap M$. If $b \in N : M$ is such that $bx = 0$ for all $x \in M$, then $bs = 0$, whence $b = 0$ since $s$ is not a divisor of 0 in $B$. On the other hand, let $f \in \mathrm{Hom}_A(M, N)$ and set $b = f(s)/s$; for all $x \in M$, there exists $t \in S$ such that $tx \in A$. Then

whence $b \in N : M$ and $f = h_b$, which proves the proposition.