Editing Pure submodule (section)

{{Short description|Module components with flexibility in module theory}}
In [[mathematics]], especially in the field of [[module theory]], the concept of '''pure submodule''' provides a generalization of [[direct summand]], a type of particularly well-behaved piece of a [[module (mathematics)|module]].  Pure modules are complementary to [[flat module]]s and generalize Prüfer's notion of [[pure subgroup]]s.  While flat modules are those modules which leave [[short exact sequence]]s exact after [[tensor product|tensoring]], a pure submodule defines a short exact sequence (known as a '''pure exact sequence''') that remains exact after tensoring with any module.  Similarly a flat module is a [[direct limit]] of [[projective module]]s, and a pure exact sequence is a direct limit of [[split exact sequence]]s.