Let $R$ be a unital ring, and let $I$ be a left $R$-module. The following conditions are equivalent.
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1. The left $R$-module $I$ is injective: for every injective left $R$-module homomorphism $\alpha: A \to B$ and every left $R$-module homomorphism $f: A \to I$, there exists a left $R$-module homomorphism $g: B \to I$ such that $g \circ \alpha = f$.
2. For every short exact sequence of left $R$-modules
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\begin{align*}
0 \longrightarrow A \xrightarrow{\iota} B \xrightarrow{\pi} C \longrightarrow 0
\end{align*}
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and every left $R$-module homomorphism $f: A \to I$, there exists a left $R$-module homomorphism $g: B \to I$ such that $g \circ \iota = f$.
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3. Every short exact sequence of left $R$-modules
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\begin{align*}
0 \longrightarrow I \xrightarrow{u} M \xrightarrow{v} N \longrightarrow 0
\end{align*}
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splits.
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4. For every left $R$-module $M$ and every injective left $R$-module homomorphism $j: I \to M$, there exists a left $R$-module homomorphism $r: M \to I$ such that $r \circ j = \operatorname{id}_I$.