Theorem: Let $A,B$ be rings and $\phi : A \to B$ be a ring homomorphism. The functor $\phi_* : B-\mathcal{M}\mathrm{od} \to A-\mathcal{M}\mathrm{od}$ between their module categories is defined by $(N,l_N) \mapsto (N,l_N \circ (\phi \otimes \mathrm{id}))$. Then the functor $\phi_*$ admits a left adjoint $\phi^* := B \otimes_A -: A-\mathcal{M}\mathrm{od} \to B-\mathcal{M}\mathrm{od}$ and a right adjoint $\phi^! := \hom_A(B,-) : A-\mathcal{M}\mathrm{od} \to B-\mathcal{M}\mathrm{od}$.