Free product of associative algebras


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, 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.

External linksEdit

  • "How to construct the coproduct of two (non-commutative) rings". Stack Exchange. January 3, 2014.