Split exact sequence

Remarks

Equivalently, there is a morphism $t \colon B \to A$ with $t \circ f = 1_A$, or there is a morphism $u \colon C \to B$ with $g \circ u = 1_C$. Every additive functor sends a split exact sequence to a split exact sequence.

Example: $0 \to \Z \xrightarrow{i} \Z \oplus \Z/2\Z \xrightarrow{p} \Z/2\Z \to 0$, with $i(n) = (n, 0)$ and $p(n, a) = a$, is split. Take $h$ to be the identity of $\Z \oplus \Z/2\Z$; then $h \circ i = i_A$ and $p_C \circ h = p$.

Non-example: $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$ is short exact but not split. A split sequence would give an isomorphism $\Z \cong \Z \oplus \Z/2\Z$, but $\Z$ has no nonzero element of finite order.