Baer's criterion

Baer's criterion: Let $R$ be a unital ring and let $Q$ be a left $R$-module. Then $Q$ is injective if and only if for every left ideal $I\subseteq R$ and every $R$-linear map $f:I\to Q$, there exists an $R$-linear map $F:R\to Q$ such that $F|_I=f$.

As a diagram, this is the injectivity diagram with $A\hookrightarrow B$ replaced by the inclusion $I\hookrightarrow R$:

This reduces checking injectivity to an extension property for module homomorphisms defined on (left) ideals of a unital ring.