---
tag: 1735VGRT2P
kind: concept
name: Tensor product of algebras
msc: [13B02, 16S10, 15A69]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_product_of_algebras
same_as:
  wikidata: Q7700713
  wikipedia: Tensor product of algebras
---

## Summary

The tensor product $A \otimes_R B$ of two [[algebras|DPBJ04JMH8]] over a commutative ring $R$ is the [[tensor product of modules|F0HQXQ0HCD]] with the multiplication $(a \otimes b)(a' \otimes b') = aa' \otimes bb'$.
For commutative algebras it is the coproduct.
A pair of algebra maps out of $A$ and $B$ to a commutative algebra is the same as one algebra map out of $A \otimes_R B$.
It describes base change, such as $\C \otimes_\R A$, and in geometry it gives the fibre product of affine schemes.

## Explanations

### L1 human

The tensor product of two algebras is the most general algebra built from them in which elements of the first commute with elements of the second.
Polynomial rings show the idea: $R[x] \otimes_R R[y]$ is $R[x, y]$.
Here $x^2 \otimes y$ corresponds to the monomial $x^2 y$.
Combining two algebras this way can also split them up: $\C \otimes_\R \C$ is isomorphic to $\C \times \C$.