# Definition:Division Ring

## Definition

A division ring is a ring with unity $\struct {R, +, \circ}$ with the following properties:

### Definition 1

$\forall x \in R^*: \exists! x^{-1} \in R^*: x^{-1} \circ x = x \circ x^{-1} = 1_R$

where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$:

$R^* = R \setminus \set {0_R}$

That is, every non-zero element of $R$ has a (unique) non-zero product inverse.

### Definition 2

Every non-zero element of $R$ is a unit.

### Definition 3

$R$ has no proper elements.

## Also known as

Some sources use this as the definition of a field, although this is non-standard: it is usually specified that a field product has to be commutative.

## Also see

• Results about division rings can be found here.