Compatibility of Quasi-Orderings and Valuations: A Baer-Krull Theorem for Quasi-Ordered Rings

摘要

In 1969, Manis introduced valuations on commutative rings. Recently the class of totally quasi-ordered rings was recently developed. In the present paper, given a quasi-ordered ring (R, ?) and a valuation v on R, we establish the notion of compatibility between v and ?, leading to a definition of the rank of (R, ?). Our main result is 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 domain to R itself. In particular, this approach generalizes to the ring case the results of the paper A Baer-Krull Theorem for Quasi-Ordered Groups (Kuhlmann and Lehericy 2017).