Free product of associative algebras

This article relies largely or entirely on a single source. Please help improve this article by introducing citations to additional sources.
Find sources: "Free product of associative algebras" – news · newspapers · books · scholar · JSTOR (May 2024)

|

Algebraic structure → Ring theory
Ring theory

Basic concepts

Rings

• Subrings
• Ideal
• Quotient ring

• Fractional ideal
• Total quotient ring
• Product ring
• Free product of associative algebras
• Tensor product of rings

Ring homomorphisms

• Kernel
• Inner automorphism
• Frobenius endomorphism

Algebraic structures

• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings

• Initial ring ℤ
• Terminal ring 0=ℤ1

Related structures

• Field

• Finite field
• Non-associative ring

• Lie ring
• Jordan ring
• Semiring

• Semifield

* * *

Commutative algebra

Commutative rings

• Integral domain

• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field

• Finite field
• Composition ring

• Polynomial ring
• Formal power series ring

Algebraic number theory

• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree

p-adic number theory and decimals

• Direct limit/Inverse limit
• Zero ring ℤ1
• Integers modulo pn ℤ/pnℤ
• Prüfer p-ring ℤ(p∞)
• Base-p circle ring 𝕋
• Base-p integers ℤ
• p-adic rationals ℤ[1/p]
• Base-p real numbers ℝ
• p-adic integers ℤp
• p-adic numbers ℚp
• p-adic solenoid 𝕋p

Algebraic geometry

• Affine variety

* * *

Noncommutative algebra

Noncommutative rings

• Division ring
• Semiprimitive ring
• Simple ring
• Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra
Operator algebra

* v
* t
* e

In algebra, the free product (coproduct) of a family of associative algebras Ai,i∈I over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the Ai'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=⨁n=1∞Tn where

T1=A⊕B,T2=(A⊗A)⊕(A⊗B)⊕(B⊗A)⊕(B⊗B),…

We then set

A*B=T/I

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

a⊗a′−aa′,b⊗b′−bb′,1A−1B.

It is then straightforward to verify that the above construction possesses the universal property of a coproduct.

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.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Free product of associative algebras. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Free_product_of_associative_algebras&oldid=3985306"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template