# Definition:Division Ring

## Definition

A **division ring** is a ring with unity $\left({R, +, \circ}\right)$ such that:

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

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

- $R^* = R \setminus \left\{ {0_R}\right\}$

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

## Alternative definitions

A **division ring** is a ring with unity $\left({R, +, \circ}\right)$ such that:

This follows from how a unit is defined.

- $(2): \quad R$ has no proper elements.

This follows from the fact that a unit is not a proper element.

- $(3): \quad R$ has no proper zero divisors.

This follows from the fact that a unit can not be a zero divisor.

## Also see

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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 55 \ (3)$