# Category:Division Rings

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This category contains results about **Division Rings**.

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.

## Subcategories

This category has the following 7 subcategories, out of 7 total.

## Pages in category "Division Rings"

The following 15 pages are in this category, out of 15 total.