---
tag: 7EN0FYFTVE
kind: concept
name: Partially ordered ring
msc: ["06F25", "16W80"]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Partially_ordered_ring
same_as:
  wikidata: Q7140405
  wikipedia: Partially ordered ring
---

## Summary

A partially ordered ring is a [[ring|0B1261B5ED]] with a [[partial order|0QV2JS9RJ3]] that is preserved by addition and for which products of nonnegative elements are nonnegative.
Rings of real functions with the pointwise order and rings of real matrices with the entrywise order are typical examples.
[[Ordered rings|H6WXC7Q71A]] are the case of a total order, and [[f-rings|VHHMZPNVEA]] are an important class of lattice-ordered rings.

## Explanations

### L1 human

Real functions on a set can be added, multiplied and compared point by point.
If $f \le g$, then $f + h \le g + h$, and the product of two nonnegative functions is nonnegative.
But two functions need not be comparable: one may be larger at some points and smaller at others.
A partially ordered ring is a ring with an order of this kind.