A weak Nullstellensatz for valuations

Abstract:

Given a real-valued pseudovaluation $p$ on a commutative ring $R$, we show how to obtain a valuation $v$ greater than or equal to $p$, and also satisfying certain upper bounds: in particular, if $p(st) = p(s) + p(t)$ for all $s,t \in S,S$ a multiplicative semigroup in $R$, then $v$ can be chosen so that $v(s) = p(s)$ for all $s \in S$.