In algebra, the free product (coproduct) of a family of associative algebras A i , i ∈ I {\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 A i {\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, T = ⨁ n = 1 ∞ T n {\displaystyle T=\bigoplus _{n=1}^{\infty }T_{n}} where
T 1 = A ⊕ B , T 2 = ( A ⊗ A ) ⊕ ( A ⊗ B ) ⊕ ( B ⊗ A ) ⊕ ( B ⊗ B ) , … {\displaystyle T_{1}=A\oplus B,\,\,T_{2}=(A\otimes A)\oplus (A\otimes B)\oplus (B\otimes A)\oplus (B\otimes B),\,\dots }
We then set
A ∗ B = T / I {\displaystyle A*B=T/I}
where I is the two-sided ideal generated by elements of the form
a ⊗ a ′ − a a ′ , b ⊗ b ′ − b b ′ , 1 A − 1 B . {\displaystyle a\otimes a'-aa',\,b\otimes b'-bb',\,1_{A}-1_{B}.}
It is then straightforward to verify that the above construction possesses the universal property of a coproduct. 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 "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.