Lemma 15.55.3. Let $R$ be a ring. Let $J$ be an $R$-module. The following are equivalent
$J$ is injective,
$\mathop{\mathrm{Ext}}\nolimits ^1_ R(M, J) = 0$ for every $R$-module $M$.
Proof. Let $0 \to M'' \to M' \to M \to 0$ be a short exact sequence of $R$-modules. Consider the long exact sequence
\[ \begin{matrix} 0 \to \mathop{\mathrm{Hom}}\nolimits _ R(M, J) \to \mathop{\mathrm{Hom}}\nolimits _ R(M', J) \to \mathop{\mathrm{Hom}}\nolimits _ R(M'', J)
\\ \phantom{0\ } \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(M, J) \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(M', J) \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(M'', J) \to \ldots
\end{matrix} \]
of Algebra, Lemma 10.71.7. Thus we see that (2) implies (1). Conversely, if $J$ is injective then the $\mathop{\mathrm{Ext}}\nolimits $-group is zero by Homology, Lemma 12.27.2 and Lemma 15.55.2. $\square$
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