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Free product of associative algebras

## Summary

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.

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

${\displaystyle T_{0}=R,\,T_{1}=A\oplus B,\,T_{2}=(A\otimes A)\oplus (A\otimes B)\oplus (B\otimes A)\oplus (B\otimes B),\,T_{3}=\cdots ,\dots }$

We then set

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

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

${\displaystyle a\otimes a'-aa',\,b\otimes b'-bb',\,1_{A}-1_{B}.}$

We then verify the universal property of coproduct holds for this (this is straightforward but we should give details.)

## 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.