In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.
Formally, we start with an algebra D over a field, and assume that D does not just consist of its zero element. 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: an associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1≠0 and every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1).
Associative division algebras
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 to isomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4).
Wedderburn's little theorem states that if D is a finite division algebra, then D is a finite field. (T. Y. Lam, A First Course in Noncommutative Rings.)
Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself of course.
Associative division algebras have no zero divisors. A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no zero divisors.
Whenever A is an associative unital 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 associative division algebra over F arises in this fashion.
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