Division algebra

In abstract algebra, a division algebra is a unitary associative algebra with 0 ≠ 1 and such that every element a has a multiplicative inverse x with ax = xa = 1.

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). 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: division ring