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The strong persistence property and symbolic strong persistence property

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Abstract

Let I be an ideal in a commutative Noetherian ring R. Then the ideal I has the strong persistence property if and only if (Ik+1: RI) = Ik for all k, and I has the symbolic strong persistence property if and only if (I(k+1): RI(1)= I(k) for all k, where I(k) denotes the kth symbolic power of I. We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial ideal has the symbolic strong persistence property.

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Acknowledgments

The authors are deeply grateful to the anonymous referee for careful reading of the manuscript, and for his/her valuable suggestions, which led to certain improvements in the quality of this paper.

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Authors and Affiliations

  1. Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran

    Mehrdad Nasernejad & Kazem Khashyarmanesh

  2. Department of Mathematics and Statistics, Queen’s University, 48 University Ave., Jeffery Hall, Kingston, Ontario, Canada, K7L 3N6

    Leslie G. Roberts

  3. Departamento de matemáticas, Centro de investigación y estudios avanzados del IPN, Apartado postal 14-740, Ciudad de México, 07000, México

    Jonathan Toledo

Authors
  1. Mehrdad Nasernejad
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  2. Kazem Khashyarmanesh
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  3. Leslie G. Roberts
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  4. Jonathan Toledo
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Corresponding author

Correspondence to Kazem Khashyarmanesh.

Additional information

Jonathan Toledo was partly supported by FORDECYT 265667.

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Nasernejad, M., Khashyarmanesh, K., Roberts, L.G. et al. The strong persistence property and symbolic strong persistence property. Czech Math J 72, 209–237 (2022). https://doi.org/10.21136/CMJ.2021.0407-20

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  • DOI: https://doi.org/10.21136/CMJ.2021.0407-20

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