Generalisations
Many statements about free modules, which are wrong for general modules over rings, are still true for certain generalisations of free modules. Projective modules are direct summands of free modules, so one can choose an injection in a free module and use the basis of this one to prove something for the projective module. Even weaker generalisations are flat modules, which still have the property that tensoring with them preserves exact sequences, and torsion-free modules. If the ring has special properties, this hierarchy may collapse, i.e. for any perfect local Dedekind ring, every torsion-free module is flat, projective and free as well.
See local ring, perfect ring and Dedekind ring.
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