Editing Projective module (section)

===Projective vs. free modules===
Any free module is projective. The converse is true in the following cases:
* if ''R'' is a field or [[skew field]]: ''any'' module is free in this case.
* if the ring ''R'' is a [[principal ideal domain]]. For example, this applies to {{nowrap|1=''R'' = '''Z'''}} (the [[integer]]s), so an [[abelian group]] is projective if and only if it is a [[free abelian group]]. The reason is that any [[submodule]] of a free module over a principal ideal domain is free.
* if the ring ''R'' is a [[local ring]]. This fact is the basis of the intuition of "locally free = projective". This fact is easy to [[mathematical proof|prove]] for [[finitely generated module|finitely generated]] projective modules. In general, it is due to {{harvtxt|Kaplansky|1958}}; see [[Kaplansky's theorem on projective modules]].

In general though, projective modules need not be free:
* Over a [[direct product of rings]] {{nowrap|''R'' × ''S''}} where ''R'' and ''S'' are [[zero ring|nonzero]] rings, both {{nowrap|''R'' × 0}} and {{nowrap|0 × ''S''}} are non-free projective modules.
* Over a [[Dedekind domain]] a non-[[principal ideal|principal]] [[ideal (ring theory)|ideal]] is always a projective module that is not a free module.
* Over a [[matrix ring]] M<sub>''n''</sub>(''R''), the natural module ''R''<sup>''n''</sup> is projective but is not free when ''n'' > 1.
* Over a [[semisimple ring]], ''every'' module is projective, but a nonzero proper left (or right) ideal is not a free module. Thus the only semisimple rings for which all projectives are free are [[division ring]]s.
The difference between free and projective modules is, in a sense, measured by the [[algebraic K-theory|algebraic ''K''-theory]] [[group (mathematics)|group]] ''K''<sub>0</sub>(''R''); see below.