Strongly AP filtrations, integral dependence and Prüferian equivalence in Dedekind domains (Q1209074)

Let \(f=(I_ n)_{n\geq 0}\) denote a filtration of ideals in a commutative ring \(A\). It is called to be approximable by powers \((AP\)- filtration) provided there is a sequence of positive integers \((k_ n)_{n\geq 0}\) such that \(I_{k_ nm}\subseteq I^ m_ n\) for all \(m,n\) and \(\lim_{n\to\infty}k_ n/n=1\). An \(AP\)-filtration is called strongly \(AP\) provided there is an integer \(r\geq 1\) such that \(I_{nr}=I^ n_ r\) for all \(n\geq 0\). The \(m\)-th truncation \(t_ mf\) of \(f\) is defined by \((I_{n+m})_{n\geq 0}\). Furthermore a filtration \(g\) is dominated by \(f\) if \(t_ mg\leq f\) for some \(m\geq 1\). The main result says that for two strongly \(AP\)-filtrations \(f,g\) in a Noetherian ring \(g\) is dominated by \(f\) if and only if \(g\) is integral over \(f\), i.e., \(R(A,g)\) is integral over \(R(A,f)\), where \(R(A,f)\) denotes the Rees ring of \(A\) with respect to \(f\). In the case of \(A\) a Nagata domain this holds without the hypothesis ``\(g\) is strongly \(AP\)''. There are further equivalent conditions concerning valuative and asymptotic equivalence of filtrations. In the particular case of a Dedekind domain there are relations to the generalized Samuel number introduced by \textit{P. Ayegnon} and \textit{D. Sangaré} in J. Pure Appl. Algebra 65, No. 1, 1-13 (1990; Zbl 0704.13016).