Editing Projective module (section)

==The category of projective modules==
Submodules of projective modules need not be projective; a ring ''R'' for which every submodule of a projective left module is projective is called [[hereditary ring|left hereditary]].

[[Quotient module|Quotients]] of projective modules also need not be projective, for example '''Z'''/''n'' is a quotient of '''Z''', but not [[torsion-free module|torsion-free]], hence not flat, and therefore not projective.

The category of finitely generated projective modules over a ring is an [[exact category]]. (See also [[algebraic K-theory]]).