In algebra, the free product (coproduct) of a family of associative algebras over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the '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.


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, where

We then set

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

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

A finite free product is defined similarly.