over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the
The notion is a ring-theoretic analog of a free product of groups.
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.