Quasi-valuations extending a valuation

Suppose F is a field with valuation v and valuation ring O-v, 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 O-v; 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 O-v 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 O-v we construct a quasi-valuation on R; we also construct a quasi-valuation on R circle times o(v) F which helps us prove our main theorem. The main theorem states that if R subset of E satisfies R boolean AND F = O-v and E is the field of fractions of R, then R and v induce a quasi-valuation w on E such that R = O-w and w extends v; thus R satisfies the properties of a quasi-valuation ring. (C) 2012 Elsevier Inc. All rights reserved.