The Stacks project

Example 47.3.6. Let $R$ be a reduced ring. Let $\mathfrak p \subset R$ be a minimal prime so that $K = R_\mathfrak p$ is a field (Algebra, Lemma 10.25.1). Then $K$ is an injective $R$-module. Namely, we have $\mathop{\mathrm{Hom}}\nolimits _ R(M, K) = \mathop{\mathrm{Hom}}\nolimits _ K(M_\mathfrak p, K)$ for any $R$-module $M$. Since localization is an exact functor and taking duals is an exact functor on $K$-vector spaces we conclude $\mathop{\mathrm{Hom}}\nolimits _ R(-, K)$ is an exact functor, i.e., $K$ is an injective $R$-module.

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