Tensor product of algebras

Summary

The tensor product $A \otimes_R B$ of two algebras over a commutative ring $R$ is the tensor product of modules 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.