Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Some Extensions of Rings with Noetherian Spectrum

  • Park, Min Ji (Department of Mathematics, College of Life Science and Nano Technology, Hannam University) ;
  • Lim, Jung Wook (Department of Mathematics, College of Natural Sciences, Kyungpook National University)
  • Received : 2019.12.20
  • Accepted : 2021.05.04
  • Published : 2021.09.30
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Abstract

In this paper, we study rings with Noetherian spectrum, rings with locally Noetherian spectrum and rings with t-locally Noetherian spectrum in terms of the polynomial ring, the Serre's conjecture ring, the Nagata ring and the t-Nagata ring. In fact, we show that a commutative ring R with identity has Noetherian spectrum if and only if the Serre's conjecture ring R[X]U has Noetherian spectrum, if and only if the Nagata ring R[X]N has Noetherian spectrum. We also prove that an integral domain D has locally Noetherian spectrum if and only if the Nagata ring D[X]N has locally Noetherian spectrum. Finally, we show that an integral domain D has t-locally Noetherian spectrum if and only if the polynomial ring D[X] has t-locally Noetherian spectrum, if and only if the t-Nagata ring $D[X]_{N_v}$ has (t-)locally Noetherian spectrum.

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Acknowledgement

The authors sincerely thank the referee for valuable comments. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2017R1C1B1008085).

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