Let \(R\) be a ring. A short exact sequence of \(R\)-modules is an exact sequence \[ 0\to A\mapsvia{i}B\mapsvia{j}C\to0 \] of \(R\)-modules and \(R\)-linear maps; equivalently, \(i\) is injective, \(j\) is surjective, and \(\im i=\ker j\) [@Hat02]. It splits if there is an \(R\)-linear map \(s\colon C\to B\) with \(j\circ s=\id_C\).
In a short exact sequence, \(j\) induces an isomorphism \(B/i(A)\cong C\).
(Splitting lemma)
For a short exact sequence \(0\to A\mapsvia{i}B\mapsvia{j}C\to0\) of \(R\)-modules, the following are equivalent:
there is an \(R\)-linear \(s\colon C\to B\) with \(j\circ s=\id_C\);
there is an \(R\)-linear \(r\colon B\to A\) with \(r\circ i=\id_A\);
there is an isomorphism \(B\cong A\oplus C\) under which \(i\) becomes \(a\mapsto(a,0)\) and \(j\) becomes \((a,c)\mapsto c\) [@Hat02].