---
tag: H6WXC7Q71A
kind: concept
name: Ordered ring
msc: ["06F25", "12J15"]
specializes: [7EN0FYFTVE]
sources:
  - url: https://en.wikipedia.org/wiki/Ordered_ring
same_as:
  wikidata: Q3782642
  wikipedia: Ordered ring
---

## Summary

An ordered ring is a commutative [[ring|0B1261B5ED]] with a [[total order|0FRXBVVR03]] that is preserved by addition and for which products of nonnegative elements are nonnegative.
The integers are the basic example, and every [[ordered field|0P3VCBDEE2]] is an ordered ring.
In an ordered ring every square is nonnegative, and the usual rules for inequalities hold.

## Explanations

### L1 human

The integers can be added, multiplied and compared, and the familiar rules link these operations.
Adding the same number to both sides of an inequality keeps it true.
The product of two nonnegative numbers is nonnegative.
An ordered ring is any ring with an order that obeys these rules, such as the integers, the rationals or the polynomials with the right order.