Flat Module

A module M over a unit ring R is called flat iff the tensor product functor - tensor _RM (or, equivalently, the tensor product functor M tensor _R-) is an exact functor.

For every R-module, M obeys the implication

which, in general, cannot be reversed.

A Z-module is flat iff it is torsion-free: hence Q and the infinite direct product Z×Z×... are flat Z-modules, but they are not projective. In fact, over a Noetherian ring or a local ring, flatness implies projectivity only for finitely generated modules. This property, together with Serre's problem, allows it to be concluded that the three above implications are equivalences if M is a finitely generated module over a polynomial ring k[X_1,...,X_n], where k is a field.

See also

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Flat Module." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FlatModule.html

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