#
nLab
totally ordered ring

### Context

#### Algebra

#### (0,1)-Category theory

# Contents

## Idea

A totally ordered ring is an ordered ring whose order forms a total order.

## Definition

This definition is adapted from Peter Freyd‘s definition of a totally ordered abelian group:

A **totally ordered ring** is an lattice-ordered ring $R$ such that for all elements $a$ in $R$, $a \leq 0$ or $-a \leq 0$.

In a totally ordered ring, the join is usually called the **maximum**, while the meet is usually called the **minimum**

If the relation $\leq$ is only a preorder, then the prelattice-ordered ring $R$ is said to be a **totally preordered ring**.

## Examples

The integers, the rational numbers, and the real numbers are totally ordered rings.

## References

- Peter Freyd,
*Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

## External links

Last revised on February 23, 2024 at 19:55:02.
See the history of this page for a list of all contributions to it.