Editing Projective module (section)

{{Short description|Direct summand of a free module (mathematics)}}
In [[mathematics]], particularly in [[algebra]], the [[Class (set theory)|class]] of '''projective modules''' enlarges the class of [[free module]]s (that is, [[module (mathematics)|module]]s with [[basis vector]]s) over a [[ring (mathematics)|ring]], keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

Every free module is a projective module, but the [[converse (logic)|converse]] fails to hold over some rings, such as [[Dedekind ring]]s that are not [[principal ideal domain]]s. However, every projective module is a free module if the ring is a principal ideal domain such as the [[integer]]s, or a (multivariate) [[polynomial ring]] over a [[field (mathematics)|field]] (this is the [[Quillen–Suslin theorem]]).

Projective modules were first introduced in 1956 in the influential book ''Homological Algebra'' by [[Henri Cartan]] and [[Samuel Eilenberg]].