# Quasi-ordered Rings

## TL;DR

This paper extends Fakhruddin's dichotomy of quasi-ordered fields to commutative rings with 1, establishing that such rings are either ordered rings or valued rings, thus unifying the treatment of these structures.

## Contribution

It introduces the concept of quasi-ordered rings and proves a similar dichotomy result as for fields, broadening the scope of Fakhruddin's work.

## Key findings

- Quasi-ordered rings are either ordered rings or valued rings.
- The paper develops a notion of quasi-ordered rings.
- The dichotomy for rings parallels that for fields.

## Abstract

A quasi-order is a binary, reflexive and transitive relation. In the Journal of Pure and Applied Algebra 45 (1987), S.M. Fakhruddin introduced the notion of (totally) quasi-ordered fields and showed that each such field is either an ordered field or else a valued field. Hence, quasi-ordered fields are very well suited to treat ordered and valued fields simultaneously. In this note, we will prove that the same dichotomy holds for commutative rings with 1 as well. For that purpose we first develop an appropriate notion of (totally) quasi-ordered rings. Our proof of the dichotomy then exploits Fakhruddin's result that was mentioned above.