Free_product_of_associative_algebras Knowpia

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 edit

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, $T=\bigoplus _{n=0}^{\infty }T_{n}$ 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'-aa',\,b\otimes b'-bb',\,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 edit

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.

External links edit

"How to construct the coproduct of two (non-commutative) rings". Stack Exchange. January 3, 2014.

Summary

Construction edit

References edit

External links edit