Free product of associative algebras

In algebra, the free product (coproduct) of a family of associative algebras $A_i,i\in I$ over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the $A_i$'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, $T=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{T}_n$ where

$T_0=R,T_1=A\oplus B,T_2=(A\otimes A)\oplus (A\otimes B)\oplus (B\otimes A)\oplus (B\otimes B),T_3=\cdots ,\dots$

We then set

$A*B=T/I$

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

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

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

A finite free product is defined similarly.

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

External links

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