division algebra

Let K be a unital ring and A a K-algebraPlanetmathPlanetmath. Defining “division” requires special considerations when the algebras are non-associative so we introduce the definition in stages.

1 Associative division algebras

If A is an associative algebra then we say A is a division algebraMathworldPlanetmath if

  1. (i)

    A is unital with identityPlanetmathPlanetmathPlanetmath 1. So for all aA,

  2. (ii)

    Also every non-zero element of A has an inverseMathworldPlanetmathPlanetmathPlanetmath. That is aA, a0, then there exists a bA such that


    We denote b by a-1 and we may prove a-1 is unique to a.

The standard examples of associative division algebras are fields, which are commutativePlanetmathPlanetmathPlanetmathPlanetmath, and the non-split quaternion algebraPlanetmathPlanetmathPlanetmath: α,βK,


where x2-α and x2-β are irreduciblePlanetmathPlanetmathPlanetmathPlanetmath over K.

2 Non-associative division algebras

For non-associative algebras A, the notion of an inverse is not immediate. We use x.y for the productPlanetmathPlanetmath of x,yA.

Invertible as endomorphismsPlanetmathPlanetmathPlanetmath: Let aA. Then define La:xa.x and Ra:xx.a. As the product of A is distributive, both La an Ra are additive endomorphisms of A. If La is invertible then we may call aleft invertible” and similarly, when Ra is invertible we may call a “right invertible” and “invertible” if both La and Ra are invertible.

In this model of invertible, A is a division algebra if, and only if, for each non-zero aA, both La and Ra invertible. Equivalently: the equations a.x=b and y.a=b have unique solutions for nonzero a,bA. However, x and y need not be equal.

A common method to produce non-associative division algebras of this sort is through Schur’s Lemma.

Invertible in the product: In some instances, the notion of invertible via endomorphisms is not sufficient. Instead, assume A has an identity, that is, an element 1A such that for all aA,


Next if aA, we say a is invertible if there exists a bA such that

a.b=1=b.a (1)

and furthermore that for all xA,

b.(a.x)=x=(x.a).b. (2)

Evidently (1) can be inferred from (2). This added assumptionPlanetmathPlanetmath substitutes for the need of associativity in the proofs of uniqueness of inverses and in solving equations with non-associative products.

Proposition 1.

If A is a finite dimensional algebra over a field, then invertible in this sense forces both La and Ra to be invertible as well.


Let xA. Then xL1=1.x=x=b.(a.x)=xLaLb. So L1=LaLb. As L1 is the identity map, La is injectivePlanetmathPlanetmath and Lb is surjective. As A is finite dimensional, injective and surjective endomorphisms are bijectiveMathworldPlanetmath. ∎

In this model, a non-associative algebra is a division algebra A if it is unital and every non-zero element is invertible.

3 Alternative division algebras

The standard examples of non-associative division algebras are actually alternative alegbras, specfically, the composition algebrasMathworldPlanetmath of fields, non-split quaternions and non-split octonions – only the latter are actually not associative. Invertible in the octonions is interpreted in the second stronger form.

Theorem 2 (Bruck-Klienfeld).

Every alternative division algebra is either associative or a non-split octonion.

This result is usually followed by two useful results which serve to omit the need to consider non-associative examples.

Theorem 3 (Artin-Zorn, Wedderburn).

A finite alternative division algebra is associative and commutative, so it is a finite field.

Theorem 4.

An alternative division algebra over an algebraically closed field is the field itself.

Title division algebra
Canonical name DivisionAlgebra
Date of creation 2013-03-22 16:52:03
Last modified on 2013-03-22 16:52:03
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 6
Author Algeboy (12884)
Entry type Definition
Classification msc 16K99
Related topic octonion
Related topic Octonion
Defines division algebra