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, 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, 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 but we should give details.)
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.
"How to construct the coproduct of two (non-commutative) rings". Stack Exchange. January 3, 2014.