---
tag: VHHMZPNVEA
kind: concept
name: f-ring
msc: ["06F25"]
specializes: [7EN0FYFTVE]
sources:
  - url: https://en.wikipedia.org/wiki/Partially_ordered_ring
---

## Summary

An f-ring is a lattice-ordered ring in which multiplying by a nonnegative element keeps disjoint elements disjoint.
The notion was introduced by Birkhoff and Pierce.
Rings of continuous real functions with the pointwise order, and products of totally ordered rings, are f-rings.
Every f-ring is a subdirect product of totally ordered rings, so f-rings behave in many ways like rings of real functions.

## Explanations

### L1 human

Two nonnegative functions are disjoint when at each point at least one of them is zero.
If you multiply one of them by any nonnegative function, it stays zero where it was zero, so the two stay disjoint.
An f-ring is a ring with a lattice order in which this property holds.
The letter f stands for function, since these rings behave like rings of functions.