Free product of associative algebras

In the category of commutative R-algebras, the free product of two algebras (in that category) is their tensor product.

Construction

We first define a free product of two algebras. Let A and B be algebras over a commutative ring R. Consider their tensor algebra, the direct sum of all possible finite tensor products of A, B; explicitly, ${\displaystyle T=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{T}_n}$ where

${\displaystyle T_0=R,T_1=A\oplus B,T_2=(A\otimes A)\oplus (A\otimes B)\oplus (B\otimes A)\oplus (B\otimes B),T_3=\cdots ,\dots }$

We then set

${\displaystyle A\ast B=T/I}$

where I is the two-sided ideal generated by elements of the form

${\displaystyle a\otimes a^{\prime }-a{a}^{\prime },b\otimes b^{\prime }-b{b}^{\prime },1_A-1_B.}$

We then verify the universal property of coproduct holds for this (this is straightforward.)

A finite free product is defined similarly.