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

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.