Faithfully Flat Module
TOPICS
A module 
over a unit ring 
is called faithfully flat if the tensor
product functor 
is exact and faithful.
A faithfully flat module is always flat and faithful, but the converse does not hold in general. For example, 
is a faithful and flat 
-module, but it is not faithfully flat: in fact 
reduces all the quotient modules 
(and the maps between them) to zero, since for all 
and all 
:
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.
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Barile, Margherita. "Faithfully Flat Module." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FaithfullyFlatModule.html