Tensor product of modules

Summary

The tensor product $M \otimes_R N$ of two modules over a commutative ring $R$ is the module that turns bilinear maps on $M \times N$ into linear maps. It is generated by the symbols $m \otimes n$, which are additive in each variable and let scalars move between the two factors. It generalises the tensor product of vector spaces, but over a ring it can be much smaller than expected, even zero. Tensor products are used for change of rings, for flatness, and for the tensor-hom adjunction.