Free product of associative algebras

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Algebraic structure → Ring theory Ring theory
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In algebra, the free product (coproduct ) of a family of associative algebras $A_{i},i\in I$ {\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

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

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

{\displaystyle T_{0}=R,,円T_{1}=A\oplus B,,円T_{2}=(A\otimes A)\oplus (A\otimes B)\oplus (B\otimes A)\oplus (B\otimes B),,円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\otimes a'-aa',,円b\otimes b'-bb',,1円_{A}-1_{B}.

{\displaystyle a\otimes a'-aa',,円b\otimes b'-bb',,1円_{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

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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.