# nLab totally ordered ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

(0,1)-category

(0,1)-topos

# 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 pseudolattice 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

## 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)