Asymptotic Primes of Ratliff-Rush Closure of Ideals with Respect to Modules

Let R be a commutative Noetherian ring, M a nonzero finitely generated R-module, and I an ideal of R. The purpose of this article is to develop the concept of Ratliff-Rush closure of I with respect to M. It is shown that the sequence , n = 1,2,..., of associated prime ideals is increasing and eventu...

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Bibliographic Details
Published inCommunications in algebra Vol. 36; no. 5; pp. 1942 - 1953
Main Authors Amjadi, Jafar, Naghipour, Reza
Format Journal Article
LanguageEnglish
Published Taylor & Francis Group 01.05.2008
Subjects
Associated primes
c-Operation
Ratliff-Rush closure
Ratliff-Rush reduction
Rees ring
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Summary:Let R be a commutative Noetherian ring, M a nonzero finitely generated R-module, and I an ideal of R. The purpose of this article is to develop the concept of Ratliff-Rush closure of I with respect to M. It is shown that the sequence , n = 1,2,..., of associated prime ideals is increasing and eventually stabilizes. This result extends Mirbagheri-Ratliff's main result in Mirbagheri and Ratliff ( 1987 ). Furthermore, if R is local, then the operation is a c*-operation on the set of ideals I of R, each ideal I has a minimal Ratliff-Rush reduction J with respect to M, and, if K is an ideal between J and I, then every minimal generating set for J extends to a minimal generating set of K.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
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ISSN:0092-7872
1532-4125
DOI:10.1080/00927870801941689