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Preprint Number 1265

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1265. Salma Kuhlman, Simon Müller
Compatibility of quasi-orderings and valuations; A Baer-Krull Theorem for quasi-ordered Rings

Submission date: 26 July 2017


In a previous paper, we introduced the class of quasi-ordered commutative rings and proved that each such ring (R,≤) is either an ordered ring or a valued ring. Here we take a further step in our investigation of this class. We develop the notion of ≤-compatible valuations, leading to a definition of the rank of (R,≤). We exploit it to establish a Baer-Krull Theorem; more precisely, fixing a valuation v on R, we describe all v-compatible quasi-orders on R.

In case where the quasi-order is an order, this yields a generalization of the classical Baer-Krull Theorem for ordered fields. Else, if we restrict attention to quasi-orders that come from valuations, our results give rise to a complete characterization of all the coarsenings, respectively all the refinements, of a given valuation v on R.

Mathematics Subject Classification: 06F25, 06A05, 13J25, 13Axx, 13A18

Keywords and phrases: Quasi-order, valuation, valued ring, ordered ring, Baer-Krull, Compatibility of orders and valuations.

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