split short exact sequence

In an abelian category, a short exact sequence$0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0$is splitif it satisfies the following equivalent conditions:

(a) there exists a homomorphism $h:C\to B$ such that $gh=1_C$;

(b) there exists a homomorphism $j:B\to A$ such that $jf=1_A$;

(c) $B$ is isomorphic to the direct sum $A\oplus C$.

In this case, we say that $h$ and $j$are backmaps or splitting backmaps.