Let $R$ be a ring, and let $A$, $B$, and $C$ be left $R$-modules. Let
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\begin{align*}
0 \longrightarrow A \xrightarrow{i} B \xrightarrow{p} C \longrightarrow 0
\end{align*}
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be a short exact sequence of left $R$-modules; that is, $i:A\to B$ and $p:B\to C$ are $R$-linear maps, $i$ is injective, $p$ is surjective, and $\operatorname{im} i=\ker p$.
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The following conditions are equivalent:
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1. There exists an $R$-[linear map](/page/Linear%20Map) $s:C\to B$ such that $p\circ s=\operatorname{id}_C$.
2. There exists an $R$-linear map $r:B\to A$ such that $r\circ i=\operatorname{id}_A$.
3. There exists an $R$-module isomorphism $\Phi:A\oplus C\to B$ compatible with the sequence, in the sense that
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\begin{align*}
\Phi(a,0)&=i(a) &&\text{for all } a\in A,\\
p(\Phi(a,c))&=c &&\text{for all } a\in A,\ c\in C.
\end{align*}