Free product of associative algebras explained

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 = o plus ^{inf ty} _{n =0} T _{n} whereT _{0} = R, T _{1} = A ⊕ B, T _{2} = (A ⊗ A) ⊕ (A ⊗ B) ⊕ (B ⊗ A) ⊕ (B ⊗ B), T _{3} = … , ...

We then set

A * B = T/I where I is the two-sided ideal generated by elements of the forma ⊗ a' - a a', b ⊗ 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

• K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with generalized identities, Section 1.4. This reference was mentioned in Web site: Coproduct in the category of (noncommutative) associative algebras . May 9, 2012 . .

External links

• Web site: How to construct the coproduct of two (non-commutative) rings . January 3, 2014 . .