Back Flachheit (Algebra) German Επίπεδο πρότυπο Greek Módulo plano Spanish مدول یکدست Persian Module plat French מודול שטוח HE Modulo piatto Italian 平坦加群 Japanese 평탄 가군 Korean Плоский модуль Russian
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences. 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 Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique.