Division algebra

In abstract algebra, a division algebra is a unitary associative algebra with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1).

Some authors omit the associativity requirement and define a division algebra to be an algebra D over a field such that for any element a in D and any non-zero element b in D there exists precisely one element x with a = bx and precisely one element y in D such that a = yb. In the remainder of this article, we will however assume associativity.

The prototypical example of a division algebra over the real numbers is given by the quaternions. Every field extension forms a division algebra over the ground field. There is no finite-dimensional division algebra over the complex numbers (except for the complex numbers themselves). The only finite-dimensional division algebras over the real numbers (up to algebra isomorphism) are:

• the real numbers themselves (dimension 1)
• the complex numbers (dimension 2)
• the quaternions (dimension 4)

Whenever A is an associative algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every division algebra over F arises in this fashion.

See also: normed division algebra, division, division ring