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Topics on strict closure of rings

Abstract

In 1971, Lipman (Am J Math 93:649–685, 1971) introduced the notion of strict closure of a ring in another, and established the underlying theory in connection with a conjecture of O. Zariski. In this paper, for further developments of the theory, we investigate three different topics related to strict closure of rings. The first one concerns construction of the closure, and the second one is the study regarding the question of whether the strict closedness is inherited under flat homomorphisms. We finally handle the question of when the Arf closure coincides with the strict closure. Examples are explored to illustrate our theorems.

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References

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Acknowledgements

The authors thank the referee for his/her careful reading, comments, and helpful suggestions.

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Correspondence to Naoki Endo.

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The first author was partially supported by JSPS Grant-in-Aid for Young Scientists 20K14299 and JSPS Overseas Research Fellowships. The second author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 21K03211. The third author was partially supported by Grant-in-Aid for JSPS Fellows 20J10517.

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Endo, N., Goto, S. & Isobe, R. Topics on strict closure of rings. Res Math Sci 8, 55 (2021). https://doi.org/10.1007/s40687-021-00292-1

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Keywords

  • Strictly closed ring
  • Arf ring
  • Weakly Arf ring

Mathematics Subject Classification

  • 13A15
  • 13B22
  • 13B30