# Definition:Division Ring

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## Definition

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

### Definition 1

- $\forall x \in R_{\ne 0_R}: \exists! x^{-1} \in R_{\ne 0_R}: x^{-1} \circ x = x \circ x^{-1} = 1_R$

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

- $R_{\ne 0_R} = R \setminus \set {0_R}$

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 **division ring** as the definition of a field.

However, this is non-standard: it is usually specified that a field product has to be commutative.

Some sources refer to a **division ring** as a **skew field**, but the latter is usually applied to a **division ring** whose ring product is specifically non-commutative.

## Also see

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

## Sources

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