Formally, we start with a non-zeroalgebraD over a field. We call D a division algebra if for any element a in D and any non-zero element b in D there exists precisely one element x in D with a = bx and precisely one element y in D such that a = yb.
For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebraif 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).
The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite-dimensional as a vector space over the reals). The Frobenius theorem states that up toisomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4).
Whenever A is an associative unital algebra over the fieldF and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in this fashion.
The center of an associative division algebra D over the field K is a field containing K. The dimension of such an algebra over its center, if finite, is a perfect square: it is equal to the square of the dimension of a maximal subfield of D over the center. 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.