Quasi-valuations extending a valuation

Abstract

Suppose F is a field with valuation v and valuation ring Ov, E is a finite field extension and w is a quasi-valuation on E extending v. We study quasi-valuations on E that extend v; in particular, their corresponding rings and their prime spectra. We prove that these ring extensions satisfy INC (incomparability), LO (lying over), and GD (going down) over Ov; in particular, they have the same Krull dimension. We also prove that every such quasi-valuation is dominated by some valuation extending v.Under the assumption that the value monoid of the quasi-valuation is a group we prove that these ring extensions satisfy GU (going up) over Ov, and a bound on the size of the prime spectrum is given. In addition, a one-to-one correspondence is obtained between exponential quasi-valuations and integrally closed quasi-valuation rings.Given R, an algebra over Ov, we construct a quasi-valuation on R; we also construct a quasi-valuation on R⊗OvF which helps us prove our main theorem. The main theorem states that if R⊆E satisfies R∩F=Ov and E is the field of fractions of R, then R and v induce a quasi-valuation w on E such that R=Ow and w extends v; thus R satisfies the properties of a quasi-valuation ring.