Ratliff–Rush filtrations associated with ideals and modules over a Noetherian ring

Let R be a commutative Noetherian ring, M a finitely generated R-module and I a proper ideal of R. In this paper we introduce and analyze some properties of r(I,M)=⋃k⩾1(Ik+1M:IkM), the Ratliff–Rush ideal associated with I and M. When M=R (or more generally when M is projective) then , the usual Ratliff–Rush ideal associated with I. If I is a regular ideal and annM=0 we show that {r(In,M)}n⩾0 is a stable I-filtration. If Mp is free for all p∈SpecR∖m-SpecR, then under mild condition on R we show that for a regular ideal I, is finite. Further if A∗(I)∩m-SpecR=∅ (here A∗(I) is the stable value of the sequence Ass(R/In)). Our generalization also helps to better understand the usual Ratliff–Rush filtration. When I is a regular m-primary ideal our techniques yield an easily computable bound for k such that for all n⩾1. For any ideal I we show that for all n≫0. This yields that is Noetherian if and only if depthM>0. Surprisingly if dimM=1 then is always a Noetherian and a Cohen–Macaulay GI(R)-module. Application to Hilbert coefficients is also discussed.