A division ring (sometimes also associative division ring) or skew field is a set , equipped with the following:
- An (infix) binary operation , called addition.
- An (infix) binary operation , called multiplication.
- Two distinct constants , called zero and one respectively.
- A unary operation denoted by the prefix symbol (Called the negative or additive inverse).
- A map
such that the following compatibility condition holds:
- forms an abelian group with group operation , identity element , and inverse operation .
- is an associative binary operation on .
- forms a group with group operation , identity element , and inverse operation . This group is called the multiplicative group and is denoted .
- We have left and right distributivity laws: and for all .
A division ring is thus like a field except that we drop the condition of commutativity of multiplication.
The term division ring is sometimes used for the more general notion of a (possibly) nonassociative division ring. Two special cases worth mentioning are power-associative division rings and alternative division rings.
Note that in some alternative conventions, the term skew field is used for the above and the term division ring is used for a skew field that is finite-dimensional as a vector space over its center (which is a field).