Corollary

Let M be a submodule of an A-module E and $\phi : E \to E/M$ the canonical homomorphism. Then ${}^t \phi$ is an isomorphism of the dual of $E/M$ onto the submodule $M'$ of $E^*$ orthogonal to $M$.

If $j : M \to E$ is the canonical injection, the kernel of ${}^t j$ is by definition the orthogonal of $M$ in $E^*$.

Moreover, in the notation of the corollary, an injective homomorphism $E^*/M' \to M^*$ is obtained from ${}^t j$ when passing to the quotient.