# Definition:Division Ring

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## 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 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.

## Sources

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.It would seem that Artin defines a field instead, although it is all very informal.If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 1944: Emil Artin and Arthur N. Milgram:
*Galois Theory*(2nd ed.) (translated by Arthur N. Milgram) ... (next): $\text I$. Linear Algebra: $\text A$. Fields