Editing Projective module (section)

=== Lifting property ===

The usual [[category theory|category theoretical]] definition is in terms of the property of [[lifting property|''lifting'']] that carries over from free to projective modules: a module ''P'' is projective [[if and only if]] for every [[surjective]] [[module homomorphism]] {{nowrap|''f'' : ''N'' ↠ ''M''}} and every module homomorphism {{nowrap|''g'' : ''P'' → ''M''}}, there exists a module homomorphism {{nowrap|''h'' : ''P'' → ''N''}} such that {{nowrap|1=''fh'' = ''g''}}. (We don't require the lifting homomorphism ''h'' to be unique; this is not a [[universal property]].)

:[[Image:Projective-module-P.svg|120px]]

The advantage of this definition of "projective" is that it can be carried out in [[category (mathematics)|categories]] more general than [[module categories]]: we don't need a notion of "free object". It can also be [[dual (category theory)|dualized]], leading to [[injective module]]s. The lifting property may also be rephrased as ''every morphism from <math>P</math> to <math>M</math> factors through every epimorphism to <math>M</math>''. Thus, by definition, projective modules are precisely the [[projective object]]s in the [[category of modules|category of ''R''-modules]].