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