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Faithfully Flat Module


A module M over a unit ring R is called faithfully flat if the tensor product functor - tensor _RM is exact and faithful.

A faithfully flat module is always flat and faithful, but the converse does not hold in general. For example, Q is a faithful and flat Z-module, but it is not faithfully flat: in fact - tensor _ZQ reduces all the quotient modules Z/nZ (and the maps between them) to zero, since for all r in Q and all a in Z/nZ:

 r tensor a=r/n tensor na=0.

See also

Faithful functor, Faithful Module, Flat Module

This entry contributed by Margherita Barile

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References

Lam, T. Y. "Flat and Faithfully Flat Modules." §4 in Lectures on Modules and Rings. New York: Springer-Verlag, pp. 122-164, 1999.

Referenced on Wolfram|Alpha

Faithfully Flat Module

Cite this as:

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

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