Remark 12.29.2. Let R \to S be a ring map. Let u : \text{Mod}_ S \to \text{Mod}_ R be u(N) = N_ R and let v : \text{Mod}_ R \to \text{Mod}_ S be v(M) = M \otimes _ R S. Then u is right adjoint to v, and u is exact and v is right exact. But conditions (a), (b), (c) of Lemma 12.29.1 do not hold in general. For example, if R = \mathbf{Z} and S = \mathbf{Z}/p\mathbf{Z}, then the injective S-module \mathbf{Z}/p\mathbf{Z} is not an injective \mathbf{Z}-module. In fact, the lemma shows all injective S-modules are injective as R-modules if and only if R \to S is a flat ring map.
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