Tensor product of algebras

The tensor product of algebras $A\otimes_R B$ of two $R$ -algebras $A,B$ over a commutative ring $R$ is the tensor product of the underlying $R$ -modules, equipped with the unique $R$ -algebra structure for which

(a\otimes b)\cdot (a'\otimes b') = (aa')\otimes (bb')

and whose structure map $R\to A\otimes_R B$ is $r\mapsto (r\cdot 1_A)\otimes 1_B = 1_A\otimes (r\cdot 1_B)$ . The multiplication is induced from the bilinear multiplication maps in $A$ and $B$ via the universal property of $\otimes_R$ .

This construction is the algebraic version of “base change” and interacts well with presentations, quotients, and localization in commutative algebra.

Examples

Over a field $k$ , one has $k[x]\otimes_k k[y]\cong k[x,y]$ as $k$ -algebras, where $k[x]$ is a polynomial ring.
If $S$ is an $R$ -algebra and $I\subseteq R$ is an ideal, then $S\otimes_R (R/I)\cong S/IS$ as $R$ -algebras.