Title: Quasi-ordered Rings,
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.