Lemma 35.4.24. If $M \in \text{Mod}_ R$ is flat, then $C(M)$ is an injective $R$-module.

**Proof.**
Let $0 \to N \to P \to Q \to 0$ be an exact sequence in $\text{Mod}_ R$. Since $M$ is flat,

is exact. By Lemma 35.4.10,

is exact. By (35.4.11.1), this last sequence can be rewritten as

Hence $C(M)$ is an injective object of $\text{Mod}_ R$. $\square$

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