Title: Compatibility of quasi-orderings and valuations; A Baer-Krull Theorem for quasi-ordered 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.