Tensor product of vector spaces

Tensor products of modules over a ring

The tensor product of two modulesA and B over a commutativeringR is defined in exactly the same way as the tensor product of vector spaces over a field: $${\displaystyle A\otimes _{R}B:=F(A\times B)/G,}$$ where now ${\displaystyle F(A\times B)}$ is the free R-module generated by the cartesian product and G is the R-module generated by these relations.