Quasi-ordered Rings
Commutative Algebra
2018-07-18 v3 Rings and Algebras
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.
Keywords
Cite
@article{arxiv.1706.04533,
title = {Quasi-ordered Rings},
author = {Simon Müller},
journal= {arXiv preprint arXiv:1706.04533},
year = {2018}
}
Comments
8 pages