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Acta Mathematica Vietnamica

, Volume 42, Issue 4, pp 605–613 | Cite as

Artinian Cofinite Modules and Going-up for \(R\subseteq \widehat {R}\)

  • Gholamreza Pirmohammadi
  • Khadijeh Ahmadi Amoli
  • Kamal BahmanpourEmail author
Article
  • 79 Downloads

Abstract

Let \((R,\operatorname {\frak m})\) be a commutative Noetherian local ring. In this paper, it is shown that the going-up theorem holds for \(R\subseteq \widehat {R}\) if and only if \(\operatorname {Rad}(I+\operatorname {Ann}_{R} A)=\operatorname {\frak m}\) for any proper ideal I of R and any non-zero Artinian I-cofinite module A. Furthermore, using the main result of Zöschinger, Arch. Math. 95, 225–231 (2010), it is shown that these equivalent conditions are equivalent to R being formal catenary with α(R) = 0 and to \(\operatorname {Att}_{R} H^{\dim M}_{I}(M)=\{\operatorname {\frak p} \in \operatorname {Assh}_{R}(M)\,:\,\operatorname {Rad}(\operatorname {\frak p}+I)=\operatorname {\frak m}\}\) for any ideal I of R and any non-zero finitely generated R-module M.

Keywords

Attached prime Cofinite module Local cohomology Noetherian ring 

Mathematics Subject Classification (2010)

13D45 14B15 13E05 

Notes

Acknowledgments

The authors are deeply grateful to the referee for his/her careful reading of the paper and valuable suggestions. Also, the authors would like to thank Professor Reza Naghipour for his careful reading of the first draft and many helpful suggestions.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  • Gholamreza Pirmohammadi
    • 1
  • Khadijeh Ahmadi Amoli
    • 1
  • Kamal Bahmanpour
    • 2
    Email author
  1. 1.Payame Noor UniversityTehranIran
  2. 2.Faculty of Sciences, Department of MathematicsUniversity of Mohaghegh ArdabiliArdabilIran

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