# Division ring

From Groupprops

## Definition

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).