Editing Projective module (section)

=== Split-exact sequences ===
A module ''P'' is projective if and only if every [[short exact sequence]] of modules of the form

:<math>0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0</math>

is a [[split exact sequence]]. That is, for every surjective module homomorphism {{nowrap|''f'' : ''B'' ↠ ''P''}} there exists a '''section map''', that is, a module homomorphism {{nowrap|''h'' : ''P'' → ''B''}} such that ''fh'' = id<sub>''P''</sub>. In that case, {{nowrap|''h''(''P'')}} is a [[direct summand]] of ''B'', ''h'' is an [[isomorphism]] from ''P'' to {{nowrap|''h''(''P'')}}, and {{nowrap|''hf''}} is a [[projection (linear algebra)|projection]] on the summand {{nowrap|''h''(''P'')}}.  Equivalently,

:<math>B = \operatorname{Im}(h) \oplus \operatorname{Ker}(f) \ \ 
\text{ where } \operatorname{Ker}(f) \cong A\ \text{ and }
\operatorname{Im}(h) \cong P.</math>