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