Formally, we start with a non-zero algebra D over a field.
For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1).
[2] Associative division algebras have no nonzero zero divisors.
Given a field F, the Brauer equivalence classes of simple (contains only trivial two-sided ideals) associative division algebras whose center is F and which are finite-dimensional over F can be turned into a group, the Brauer group of the field F. One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebras (see also quaternions).
For infinite-dimensional associative division algebras, the most important cases are those where the space has some reasonable topology.
For a non-associative example, consider the complex numbers with multiplication defined by taking the complex conjugate of the usual multiplication: This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.
There are infinitely many other non-isomorphic commutative, non-associative, finite-dimensional real divisional algebras, but they all have dimension 2.
In fact, every finite-dimensional real commutative division algebra is either 1- or 2-dimensional.
Dropping the requirement of commutativity, Hopf generalized his result: Any finite-dimensional real division algebra must have dimension a power of 2.
Later work showed that in fact, any finite-dimensional real division algebra must be of dimension 1, 2, 4, or 8.
This was independently proved by Michel Kervaire and John Milnor in 1958, again using techniques of algebraic topology, in particular K-theory.
The challenge of constructing a division algebra of three dimensions was tackled by several early mathematicians.
On the other hand, we can construct a division algebra without multiplicative inverses by taking the quaternions and modifying the product, setting
while leaving the rest of the multiplication table unchanged.