Asymptotic closure of filtrations on rings with Krull dimension 1 (Q1383601)

In the present paper, the concept of asymptotic closure \(\overline {(f,M)}\) of a filtration \(f\) relative to a module \(M\) is introduced and investigated. The methods used by the authors in previous notes on integral and prüferian closure operations of a filtration prove to be efficient here, despite the complexity of the asymptotic closure operation comparatively to the integral and prüferian closure operation. Our main result gives a complete description of the asymptotic closure \(\overline f\) of a filtration \(f\) on a Dedekind ring \(A\), in terms of the prime ideals of which \(\sqrt f\) is the product, when \(f\) belongs to some class containing noetherian filtrations.