Free product of 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.

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, ${\displaystyle T=\bigoplus _{n=0}^{\infty }T_{n}}$ where

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