High Quality Content by WIKIPEDIA articles! In abstract algebra, a flat module over a ring R is an R-module M such that 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. Vector spaces over a field are flat modules. Free modules, or more generally projective modules, are also flat, over any R. For finitely generated modules over a Noetherian local ring, flatness, projectivity, and freeness are all equivalent. In commutative algebra, and more generally in algebraic geometry, flatness has come to play a major role since Serre's paper Geometrie Algebrique et Geometrie Analytique. See also flat morphism.

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