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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