Article
Kyungpook Mathematical Journal 2021; 61(3): 487-494
Published online September 30, 2021
Copyright © Kyungpook Mathematical Journal.
Some Extensions of Rings with Noetherian Spectrum
Min Ji Park, Jung Wook Lim*
Department of Mathematics, College of Life Science and Nano Technology, Hannam University, Daejeon, 34430, Republic of Korea
e-mail : mjpark5764@gmail.com
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu, 41566, Republic of Korea
e-mail : jwlim@knu.ac.kr
Received: December 20, 2019; Revised: April 29, 2020; Accepted: May 4, 2020
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]Nv has (t-)locally Noetherian spectrum.
Keywords: radically finite ideal, Noetherian spectrum, (t-)locally Noetherian spectrum, Serre's conjecture ring, (t-)Nagata ring, finite (t-)character.