Tensor product of commutative algebras
= Tensor product of commutative algebras
{title2=$A\otimes_RB$}
For commutative $R$-algebras, their module <tensor product> becomes a commutative algebra by $(a\otimes b)(a'\otimes b')=aa'\otimes bb'$. Its <universal property> is that pairs of compatible $R$-algebra maps from $A$ and $B$ into a commutative algebra $C$ correspond to maps $A\otimes_RB\to C$. This coproduct of algebras gives, contravariantly, the <fibre product of schemes> on affine charts.