Abstract
A right R-module P R is called flat if P ⊗ R — is an exact functor on \( {}_R\mathfrak{M} \), the category of left R-modules. Specifically, this requires that, if A → B is injective in \( {}_R\mathfrak{M} \), then P ⊗ R A → P ⊗ R B is injective also. Projective modules are flat, but flat modules enjoy an important property not shared by projective modules: they are closed w.r.t. direct limits. In particular, a module is flat if all f.g. submodules are flat. Over ℤ, the flat modules are just the torsion-free abelian groups.
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