Yes, it comes for free. A short exact sequence of *abelian* groups, or modules, splits iff it cosplits iff the middle term is the sum of the other two terms (sorry if I misread and you already know this!). The key here is that maps of modules and be added/subtracted: Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1<sub>B</sub>-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. Together the splitting and cosplitting exhibit B as the direct sum of A and C. A dual trick shows that a cosplit sequences are split: if p:B→A is a cosplitting, then (1<sub>B</sub>-fp):B→B is a projection onto a submodule of itself which is isomorphic to C via g, i.e. a cosplitting, so again B is the sum of A and C via these maps. More generally, this works in any abelian category.