# Free product of associative algebras

In algebra, the free product (coproduct) of a family of associative algebras $\displaystyle{ A_i, i \in I }$ over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the $\displaystyle{ A_i }$'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 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, $\displaystyle{ T = \bigoplus_{n=0}^{\infty} T_n }$ where

$\displaystyle{ T_0 = R, \, T_1 = A \oplus B, \, T_2 = (A \otimes A) \oplus (A \otimes B) \oplus (B \otimes A) \oplus (B \otimes B), \, T_3 = \cdots, \dots }$

We then set

$\displaystyle{ A * B = T/I }$

where I is the two-sided ideal generated by elements of the form

$\displaystyle{ a \otimes a' - a a', \, b \otimes b' - bb', \, 1_A - 1_B. }$

We then verify the universal property of coproduct holds for this (this is straightforward.)

A finite free product is defined similarly.

## References

• K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with generalized identities, Section 1.4. This reference was mentioned in