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Free product of associative algebras

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Algebraic structure → Ring theory
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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 = 0 ∞ T n {\displaystyle T=\bigoplus _{n=0}^{\infty }T_{n}} where

T 0 = R , T 1 = A ⊕ B , T 2 = ( A ⊗ A ) ⊕ ( A ⊗ B ) ⊕ ( B ⊗ A ) ⊕ ( B ⊗ B ) , T 3 = ⋯ , … {\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

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}.}

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

A finite free product is defined similarly.

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