A skew field, also known as a division ring, is a (not necessarily commutative) ring in which every element has a two-sided inverse. Equivalently, a skew field is a field in which multiplication does not necessarily commute. That is, it is a set along with two operations, and such that:
- There are elements such that and for all . (Existence of additive and multiplicative identities.)
- For each other than 0, there exist elements such that and . (Existence of additive and multiplicative inverses.)
- for all (Commutativity of addition.)
- for all (Associativity of addition.)
- (Associativity of multiplication.)
- and (The distributive property.)
Every field is a skew field. The most famous example of a skew field that is not also a field is the collection of quaternions.