Let \(R\) be a ring and let \(\xi\colon 0 \to A \mapsvia{d_1} B \mapsvia{d_2} C \to 0\) be a short exact sequence of \(R\)-modules. The following are equivalent:
\(\xi\) admits a right-splitting: an \(R\)-linear \(s\colon C\to B\) with \(d_2\circ s=\id_C\).
\(\xi\) admits a left-splitting: an \(R\)-linear \(t\colon B\to A\) with \(t\circ d_1=\id_A\).
There is an isomorphism \(\varphi\colon B\to A\oplus C\) with \(\varphi\circ d_1=\iota_A\) and \(\pi_C\circ\varphi=d_2\), where \(\iota_A\colon A\to A\oplus C\) is the inclusion and \(\pi_C\colon A\oplus C\to C\) is the projection.
A projective or injective module characterizes splitting for all sequences at once: \(C\) is projective if and only if every short exact sequence of \(R\)-modules \(0\to A'\to B'\to C\to 0\) splits, and \(A\) is injective if and only if every short exact sequence \(0\to A\to B'\to C'\to 0\) splits.
A single short exact sequence can split with \(C\) not projective. Over \(\ZZ\), the sequence \(0\to \ZZ/2\ZZ \to \ZZ/2\ZZ \oplus \ZZ/2\ZZ \to \ZZ/2\ZZ\to 0\) with the inclusion of the first summand and the projection onto the second splits, but \(\ZZ/2\ZZ\) is not a projective \(\ZZ\)-module, since the surjection \(\ZZ\to\ZZ/2\ZZ\) has no section.