The structure of valuation rings

Abstract

Abstract A valuation ν of a ring R has a unique extension νc to a valuation of its quotient ring Q = Qc(R), called the quotient valuation of ν. The relationship between the valuation ring Rν of localization of Rν at the valuation prime ideal Pν whenever, e.g. Rν 9 Pν ⊆ R∗ where R ∗ is the set or regular elements of R. This happens, in particular, when the quotient valuation ring is a local ring, for example, whenever R is a domain. We also study the question: which valuations of Q=Qc(R) induce valuations in R? A sufficient conditions is the existence of a denominator set S of R in Q such that w(s) = 0 Λsϵ S, where w is a valuation of Q, and Q=RS−1={rs−1|rϵR, sϵS}. Then S is a denominator set of 0w-values (i.e., zero w-values). In this case, the valuation ring of w is Qw=RwS−1 where w is the induced valuation. Again, a sufficient condition is that Qw is local, in which case Qw is the local ring of R w at P w . These topics are taken up in §1. The remainder of the paper is devoted to proofs of theorems merely stated in [1], or had proofs only sketched there.