Proposition 11

Let M, N be two sub-A-modules of B. Suppose that M is invertible. Then: (i) The canonical homomorphism $M \otimes_A N \to M . N$ is bijective. (ii) $N : M = N . (A : M)$ and $N = (N : M) . M$.

Let $j$ be the canonical injection $N \to B$. Since $M$ is a flat $A$-module (Corollary to Theorem 4), $1 \otimes j : M \otimes_A N \to M \otimes_A B$ is injective. But, as $B = S^{-1}A$, the $B$-module $M \otimes_A B$ is equal to $S^{-1}M$ and hence is identified with $B$ since $M$ is non-degenerate (no. 5, Proposition 8). If this identification is made, the image of $1 \otimes j$ is $M . N$, whence (i).

Let us set $M' = A : M$. Then obviously $M'.N \subset N : M$ and $M.(N : M) \subset N$. On the other hand, since $M.M' = A$ (Proposition 10),

and $N = M.M'.N \subset M.(N : M)$, whence (ii).