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

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.