tensor product of modules

In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (roughly speaking, "multiplication") to be carried out in terms of linear maps (module homomorphisms).

For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.

R-Mod, the category of modules over a commutative ring R, is a monoidal category with the tensor product of modules serving as the monoidal product and the ring R (thought of as a module over itself) serving as the unit.