Free product of associative algebras

In algebra, the free product (coproduct) of a family of associative algebras [math]\displaystyle{ A_i, i \in I }[/math] over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the [math]\displaystyle{ A_i }[/math]'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, [math]\displaystyle{ T = \bigoplus_{n=0}^{\infty} T_n }[/math] where

[math]\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 }[/math]

We then set

[math]\displaystyle{ A * B = T/I }[/math]

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

[math]\displaystyle{ a \otimes a' - a a', \, b \otimes b' - bb', \, 1_A - 1_B. }[/math]

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.