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Let I be an ideal in a Noetherian ring R and let \(\widetilde{I}\) be its Ratliff-Rush closure. In this paper we study the asymptotic Ratliff-Rush number, i.e. \(h(I)=\min \{n\in \mathbb N_+ \mid I^m=\widetilde{I^m}, \ \forall \ m\ge n\}\) , in the one-dimensional case. Since \(1\le h(I)\le r(I)\) , where r(I) is the reduction number of I, we look for conditions that determine the extremal values of h(I).

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On the Ratliff-Rush Closure of an Ideal of a One-dimensional Ring

  • 会议论文

  • 发表日期：

    2025年

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  • Veronica Crispin Quinonez,
  • Marco D’Anna,
  • Vincenzo Micale

摘要

Let I be an ideal in a Noetherian ring R and let \(\widetilde{I}\) be its Ratliff-Rush closure. In this paper we study the asymptotic Ratliff-Rush number, i.e. \(h(I)=\min \{n\in \mathbb N_+ \mid I^m=\widetilde{I^m}, \ \forall \ m\ge n\}\) , in the one-dimensional case. Since \(1\le h(I)\le r(I)\) , where r(I) is the reduction number of I, we look for conditions that determine the extremal values of h(I).