Editing Flat module (section)

{{short description|Algebraic structure in ring theory}}
In [[algebra]], '''flat modules''' include [[free module]]s, [[projective module]]s, and, over a [[principal ideal domain]], [[torsion-free module]]s. Formally, a [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'' is ''flat'' if  taking the [[tensor product of modules|tensor product]] over ''R'' with ''M'' preserves [[exact sequence]]s. A module is '''faithfully flat''' if taking the tensor product with a sequence produces an exact sequence [[if and only if]] the original sequence is exact.

Flatness was introduced by {{harvs|txt|authorlink=Jean-Pierre Serre|last=Serre|first=Jean-Pierre|year=1956}} in his paper ''[[Géometrie Algébrique et Géométrie Analytique]]''.