Theorem 2 (Morita)

a) Let V and W be left B-modules. The mapping $g\mapsto 1_P\otimes g$ is a bijection from Hom$_B(V,W)$ to Hom$_A(P\otimes_BV,P\otimes_BW)$. The inverse bijection sends an element $h$ of Hom$_A(P\otimes_BV,P\otimes_BW)$ to the element $\theta_W\circ (1_Q\otimes h)\circ \theta_V^{-1}$ of Hom$_B(V,W)$.

b) Every left A-module M is isomorphic to a module of the form $P\otimes_BV$, where V is a left B-module.

Let V and W be left B-modules. By Lemma 1 of VIII, p. 101, the mapping $\varphi :g\mapsto 1_P\otimes g$ from Hom$_B(V,W)$ to Hom$_A(P\otimes_BV,P\otimes_BW)$ is injective. By interchanging the roles of P and Q (and of A and B), we see that the mapping $\psi :h\mapsto 1_Q\otimes h$ from Hom$_A(P\otimes_BV,P\otimes_BW)$ to Hom$_A(Q\otimes_AP\otimes_BV,Q\otimes_AP\otimes_BW)$ is also injective. Now, the composition $\psi \circ \varphi$ is the mapping $g\mapsto \theta^{-1}_W\circ g\circ \theta_V$. It is bijective, hence so is $\psi$. Consequently, $\varphi$ is bijective, and its inverse is the mapping $h\mapsto \theta_W\circ (1_Q\otimes h)\circ \theta^{-1}_V$.

Assertion b) follows from the fact that $\lambda_M$ is an isomorphism from $P\otimes_B$ $\otimes_A\otimes M$ to M.

Let V be a left B-module and W a submodule of V. Since the B-module P is projective (VIII, p. 101, Theorem 1), the canonical mapping from $P\otimes_BW$ to $P\otimes_BV$ is injective. We identify $P\otimes_BW$ with its image in $P\otimes_BV$ through this mapping. We adopt the analogous conventions when P and B are replaced by Q and A.