Free product of associative algebras

In algebra, the free product (coproduct) of a family of associative algebras $$A_i, i \in I$$ over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the $$A_i$$'s. The free product of two algebras A, B is denoted by A&thinsp;∗&thinsp;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, $$T = \bigoplus_{n=0}^{\infty} T_n$$ where
 * $$T_0 = R, \, T_1 = A \oplus B, \, T_2 = (A \otimes A) \oplus (A \otimes B) \oplus (B \otimes A) \oplus (B \otimes B), \, T_3 = \cdots, \dots$$

We then set
 * $$A * B = T/I$$

where I is the two-sided ideal generated by elements of the form
 * $$a \otimes a' - a a', \, b \otimes b' - bb', \, 1_A - 1_B.$$

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

A finite free product is defined similarly.