Submission date: 20 June 2017

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.

Mathematics Subject Classification: 13Axx, 06Fxx

Keywords and phrases: Quasi-orders, valued rings, ordered rings.

Full text arXiv 1706.04533: pdf, ps.