# Free product of associative algebras

In algebra, the free product (coproduct) of a family of associative algebras $A_{i},i\in I$ over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the $A_{i}$ 's. The free product of two algebras A, B is denoted by AB. The notion is a ring-theoretic analog of a free product of groups.

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, B be two algebras over a commutative ring R. Consider their tensor algebra, the direct sum of all possible finite tensor products of A, B; explicitly, $T=\bigoplus _{n=0}^{\infty }T_{n}$ where

$T_{0}=R,\,T_{1}=A\oplus B,\,T_{2}=(A\otimes A)\oplus (A\otimes B)\oplus (B\otimes A)\oplus (B\otimes B),\,T_{3}=\cdots ,\dots$ We then set

$A*B=T/I$ where I is the two-sided ideal generated by elements of the form

$a\otimes a'-aa',\,b\otimes b'-bb',\,1_{A}-1_{B}.$ We then verify the universal property of coproduct holds for this (this is straightforward but we should give details.)