A module M over a unit ring R is called faithfully flat if the tensor product functor -⊗_R M 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 -⊗_Z Q reduces all the quotient modules Z/n Z (and the maps between them) to zero, since for all r element Q and all a element Z/n Z: r⊗a = r/n ⊗n a = 0.

faithful functor | faithful module | flat module