Tensor Product of Modules - Relationship To Flat Modules

Relationship To Flat Modules

In general, is a bifunctor which accepts a right and a left R module pair as input, and assigns them to the tensor product in the category of abelian groups.

By fixing a right R module M, a functor arises, and symmetrically a left R module N could be fixed to create a functor . Unlike the Hom bifunctor, the tensor functor is covariant in both inputs.

It can be shown that M⊗- and -⊗N are always right exact functors, but not necessarily left exact. By definition, a module T is a flat module if T⊗- is an exact functor.

If {m_i}_i∈I and {n_j}_j∈J are generating sets for M and N, respectively, then {m_i⊗n_j}_i∈I,j∈J will be a generating set for M⊗N. Because the tensor functor M⊗_R- sometimes fails to be left exact, this may not be a minimal generating set, even if the original generating sets are minimal.

When the tensor products are taken over a field F so that -⊗- is exact in both positions, and the generating sets are bases of M and N, it is true that indeed forms a basis for M⊗_F N.