Definition Flat module

general rings

when r isn t commutative 1 needs more careful statement that, if m flat left r-module, tensor product m maps exact sequences of right r-modules exact sequences of abelian groups.

taking tensor products (over arbitrary rings) right exact functor. therefore, r-module m flat if , if injective homomorphism k → l of r-modules, induced homomorphism k



⊗


{\displaystyle \otimes }

m → l



⊗


{\displaystyle \otimes }

m injective.