For every module M over a unit ring R, the tensor product functor -⊗_R M is a covariant functor from the category of R-modules to itself. It maps every R-module N to N⊗_R M and every module homomorphism f:N⟶P to the module homomorphism f⊗1:N⊗_R M⟶P⊗_R M defined by f⊗1(n⊗m) = f(n)⊗m. The tensor product functor M⊗_R - is defined similarly.

exact functor | faithfully flat module | flat module