Free product of associative algebras

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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 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, ${\displaystyle T={\displaystyle \underset{n=1}{\overset{\mathrm{\infty }}{⨁}}}{T}_n}$ where

${\displaystyle T_1=A\oplus B,T_2=(A\otimes A)\oplus (A\otimes B)\oplus (B\otimes A)\oplus (B\otimes B),\dots }$

We then set

${\displaystyle A*B=T/I}$

where I is the two-sided ideal generated by elements of the form

${\displaystyle a\otimes a^{\prime }-a{a}^{\prime },b\otimes b^{\prime }-b{b}^{\prime },1_A-1_B.}$

It is then straightforward to verify that the above construction possesses the universal property of a coproduct.

A finite free product is defined similarly.

• 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. https://math.stackexchange.com/q/1337405. 

• "How to construct the coproduct of two (non-commutative) rings". Stack Exchange. January 3, 2014. https://math.stackexchange.com/q/625874. 

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