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Title:  RatliffRush filtrations associated with ideals and modules over a Noetherian ring 
Authors:  PUTHENPURAKAL, TJ ZULFEQARR, F 
Keywords:  reductions 
Issue Date:  2007 
Publisher:  ACADEMIC PRESS INC ELSEVIER SCIENCE 
Citation:  JOURNAL OF ALGEBRA, 311(2), 551583 
Abstract:  Let R be a commutative Noetherian ring, M a finitely generated Rmodule and I a proper ideal of R. In this paper we introduce and analyze some properties of r(I, M) = Uk >= 1 (Ik + 1 M : Ik M), the RatliffRush ideal associated with I and M. When M = R (or more generally when M is projective) then r(I, M) = (I) over tilde, the usual RatliffRush ideal associated with I. If I is a regular ideal and ann M = 0 we show that {r(In, M)}(n >= 0) is a stable Ifiltration. If Mp is free for all p is an element of Spec R \ mSpec R, then under mild condition on R we show that for a regular ideal I, l(r(I, M)/(I) over tilde) is finite. Further r(I, M) = (I) over tilde if A*(I) boolean AND mSpec R = 0 (here A*(I) is the stable value of the sequence Ass(R / In)). Our generalization also helps to better understand the usual RatliffRush filtration. When I is a regular mprimary ideal our techniques yield an easily computable bound for k such that (I) over tilde (n) = (In + k : Ik) for all n >= 1. For any ideal I we show that (In M) over tilde = In M + H1(0)(M) for all n >> 0. This yields that (R) over tilde (I, M) = circle times(n >= 0) In M is Noetherian if and only if depth M > 0. Surprisingly if dim M = 1 then (G) over tilde (I)(M) = circle times(n >= 0) (In M) over tilde / In + 1 M is always a Noetherian and a CohenMacaulay G(I) (R)module. Application to Hilbert coefficients is also discussed. (c) 2007 
URI:  http://dx.doi.org/10.1016/j.jalgebra.2007.01.006 http://dspace.library.iitb.ac.in/xmlui/handle/10054/3388 http://hdl.handle.net/10054/3388 
ISSN:  00218693 
Appears in Collections:  Article

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