Quasi-valuations and algebras over valuation domains

Abstract

Suppose F is a field with valuation v and valuation domain Ov, and R is an Ov-algebra. We prove that R satisfies SGB (strong going between) over Ov. We give a necessary and sufficient condition for R to satisfy LO (lying over) over Ov. Using the filter quasi-valuation constructed in [Sarussi, S. (2012). Quasi-valuations extending a valuation. J. Algebra. 372:318–364], we show that if R is torsion-free over Ov, then R satisfies GD (going down) over Ov. In particular, if R is torsion-free and , then for any chain in there exists a chain in covering it. Assuming R is torsion-free over Ov and , we prove that R satisfies INC (incomparabilty) over Ov. Assuming in addition that , we deduce that R and Ov have the same Krull dimension and a bound on the size of the prime spectrum of R is given. Under certain assumptions on R and a quasi-valuation defined on it, we prove that the quasi-valuation ring satisfies GU (going up) over Ov. Combining these five properties together, we deduce that any maximal chain of prime ideals of the quasi-valuation ring is lying over , in a one-to-one correspondence.