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 (I k+1 M : I k M), the RatliffRush ideal associated with I and M. When M = R (or more generally when M is projective) then r(I, M) = I , the usual Ratliff-Rush ideal associated with I . If I is a regular ideal and ann M = 0 we show that {r(I n , M)} n 0 is a stable I -filtration. If M p is free for all p ∈ Spec R \ m-Spec R, then under mild condition on R we show that for a regular ideal I , (r(I, M)/ I ) is finite. Further r(I, M) = I if A * (I ) ∩ m-Spec R = ∅ (here A * (I ) is the stable value of the sequence Ass(R/I n )). 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 I n = (I n+k : I k ) for all n 1. For any ideal I we show that I n M = I n M + H 0 I (M) for all n 0. This yields that R(I, M) = n 0 I n M is Noetherian if and only if depth M > 0. Surprisingly if dim M = 1 then G I (M) = n 0 I n M/ I n+1 M is always a Noetherian and a Cohen-Macaulay G I (R)-module. Application to Hilbert coefficients is also discussed.

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Let R be a Cohen-Macaulay local ring with a canonical module R . Let I be an -primary ideal of R and M, a maximal Cohen-Macaulay R-module. We call the function n −→ Hom R M R /I n+1 R the dual Hilbert-Samuel function of M with respect to I. By a result of Theodorescu, this function is of polynomial type. We study its first two normalized coefficients. In particular, we analyze the case when R is Gorenstein.

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