Proposition 5

Let V be a left B-module. The mapping $W\mapsto P\otimes_BW$ is an isomorphism from the set of B-submodules of V, ordered by inclusion, to the set of A-submodules of $P\otimes_BV$, ordered by inclusion. The inverse isomorphism sends an A-submodule N of $P\otimes_BV$ to the image under $\theta_V$ of the B-submodule $Q\otimes_AN$ of $Q\otimes_AP\otimes_BV$.

Denote by $D_B(V)$ the set of B-submodules of V, ordered by inclusion, and define the sets $D_A(P\otimes_BV)$ and $D_B(Q\otimes_AP\otimes_BV)$ likewise. Let $\varphi : D_B(V)\rightarrow$ $D_A(P\otimes_BV)$ be the mapping $W\mapsto P\otimes_BW$ and $\psi$ the mapping from $D_A(P\otimes_BV)$ to $D_B(Q\otimes_AP\otimes_BV)$ given by $N\mapsto Q\otimes_AN$. These are increasing mappings, and the composition $\psi \circ \varphi$ is the mapping $W\mapsto \theta^{-1}_V$(W), which is bijective. Consequently, $\varphi$ is injective, and $\psi$ is surjective. By replacing B with A and V with $P\otimes_BV$, we see that $\psi$ is also injective. Hence, $\varphi$ and $\psi$ are bijective, and the inverse mapping of $\varphi$ is indeed that described in the proposition.