# Definition:Division Ring/Definition 1

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## Definition

A **division ring** is a ring with unity $\struct {R, +, \circ}$ 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 $0_R$:

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

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

## Also see

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

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**division ring** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**division ring**

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- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 55 \ (3)$. The definition of a ring and its elementary consequences