Compatibility of quasi-orderings and valuations; A Baer-Krull Theorem for quasi-ordered Rings

TL;DR

This paper extends valuation theory to quasi-ordered rings by defining compatibility and establishing a Baer-Krull type theorem, linking valuations and quasi-orders in a new algebraic framework.

Contribution

It introduces the concept of compatibility between valuations and quasi-orders and proves a Baer-Krull theorem for quasi-ordered rings, generalizing existing valuation results.

Findings

Defined compatibility between valuations and quasi-orders.

Proved a Baer-Krull theorem for quasi-ordered rings.

Characterized v-compatible quasi-orders via residue class rings.

Abstract

In his work of 1969, Merle E. Manis introduced valuations on commutative rings. Recently, the class of totally quasi-ordered rings was developped by the second author. In the present paper, we establish the notion of compatibility between valuations and quasi-orders on rings, leading to a definition of the rank of a quasi-ordered ring. Moreover, we prove a Baer-Krull Theorem for quasi-ordered rings: fixing a Manis valuation v on R, we characterize all v-compatible quasi-orders of R by lifting the quasi-orders from the residue class ring to R itself.