Corollary

If $M$ and $N$ are two non-degenerate sub-A-modules of $B$, the A-modules $M + N, M.N$ and $M \cap N$ are non-degenerate.

The assertion is trivial for $M + N$; on the other hand if $s \in S \cap M$ and $t \in S \cap N$, then $st \in S \cap (M.N)$ and $st \in S \cap (M \cap N)$.

Given two sub-A-modules $M$ and $N$ of $B$, let us denote by $N : M$ the sub-A-module of $B$ consisting of those $b \in B$ such that $bM \subset N$ (Chapter I, $§ 2$, no. 10, Remark). If every $b \in N : M$ is mapped to the homomorphism $h_b : x \mapsto bx$ of $M$ to $N$, a canonical homomorphism $b \mapsto h_b$ is obtained from $N : M$ to $\operatorname{Hom}_A(M, N)$.