# 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.

## Construction

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.)

## References

- 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 links

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