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 A ∗ B. 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 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, $T=\bigoplus _{n=0}^{\infty }T_{n)$ 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.)

A finite free product is defined similarly.

• K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with generalized identities, Section 1.4. This reference was mentioned in "Coproduct in the category of (noncommutative) associative algebras". Stack Exchange. May 9, 2012.