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Paper   IPM / M / 8293
School of Mathematics
  Title:   Asymptotic primes of Ratliff-Rush closure of ideals with respect to modules
  Author(s):  R. Naghipour (Joint with J. Amjadi)
  Status:   Published
  Journal: Comm. Algebra
  Vol.:  36
  Year:  2008
  Pages:   1942-1953
  Supported by:  IPM
Let R be a commutative Noetherian ring, M a non-zero finitely generated R-module and I an ideal of R. The purpose of this paper is to develop the concept of Ratliff-Rush closure ~I(M) of I with respect to M. It is shown that the sequence AssRR/~In(m), n−1,2,..., of associated prime ideals is increasingly an eventually stabilizes. This result extends Mirbagheri-Ratliff's main result in [10]. Furthermore, if R is local, then the operation I~I(M) 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.

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