Free product of associative algebras

In algebra, the free product (coproduct) of a family of associative algebras formula_1 over a commutative ring "R" is the associative algebra over "R" that is, roughly, defined by the generators and the relations of the formula_2'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, formula_3 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.)