An R-module M is flat iff the tensor functor −⊗ᵣM preserves short exact sequences (equivalently, Tor₁^R(−, M) vanishes). Over ℤ: flat ⇔ torsion-free (Lazard: flat ⇔ filtered colimit of free modules).
An R-module M is flat iff the tensor functor −⊗ᵣM preserves short exact sequences (equivalently, Tor₁^R(−, M) vanishes). Over ℤ: flat ⇔ torsion-free (Lazard: flat ⇔ filtered colimit of free modules).