# 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

### Definition 3

- $R$ has no proper elements.

## Also see

- Results about
**division rings**can be found here.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields