Let $R$ be a ring, and let

\begin{align*} 0 \longrightarrow A \xrightarrow{i} B \xrightarrow{p} C \longrightarrow 0 \end{align*}

be a short exact sequence of left $R$-modules. Thus $i: A \to B$ and $p: B \to C$ are $R$-module homomorphisms, $i$ is injective, $p$ is surjective, and $\operatorname{im} i = \ker p$.

The following conditions are equivalent:

1. The short exact sequence is split, meaning that there exists an $R$-module isomorphism $\Phi: A \oplus C \to B$ such that $\Phi(a,0)=i(a)$ for every $a \in A$ and $p(\Phi(a,c))=c$ for every $(a,c) \in A \oplus C$. 2. There exists an $R$-module homomorphism $s: C \to B$ such that $p \circ s = \operatorname{id}_C$. 3. There exists an $R$-module homomorphism $r: B \to A$ such that $r \circ i = \operatorname{id}_A$.