[Theorem 1, Baer]

Lemma 19.2.6 (Baer's criterion). Let $R$ be a ring. An $R$-module $Q$ is injective if and only if in every commutative diagram

$\xymatrix{ \mathfrak {a} \ar[d] \ar[r] & Q \\ R \ar@{-->}[ru] }$

for $\mathfrak {a} \subset R$ an ideal, the dotted arrow exists.

Proof. This is the equivalence of (1) and (3) in More on Algebra, Lemma 15.55.4; please observe that the proof given there is elementary (and does not use $\text{Ext}$ groups or the existence of injectives or projectives in the category of $R$-modules). $\square$

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