• Title/Summary/Keyword: t-locally Noetherian

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A Note on S-Noetherian Domains

  • LIM, JUNG WOOK
    • Kyungpook Mathematical Journal
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    • v.55 no.3
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    • pp.507-514
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    • 2015
  • Let D be an integral domain, t be the so-called t-operation on D, and S be a (not necessarily saturated) multiplicative subset of D. In this paper, we study the Nagata ring of S-Noetherian domains and locally S-Noetherian domains. We also investigate the t-Nagata ring of t-locally S-Noetherian domains. In fact, we show that if S is an anti-archimedean subset of D, then D is an S-Noetherian domain (respectively, locally S-Noetherian domain) if and only if the Nagata ring $D[X]_N$ is an S-Noetherian domain (respectively, locally S-Noetherian domain). We also prove that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D[X] is a t-locally S-Noetherian domain, if and only if the t-Nagata ring $D[X]_{N_v}$ is a t-locally S-Noetherian domain.

Some Extensions of Rings with Noetherian Spectrum

  • Park, Min Ji;Lim, Jung Wook
    • Kyungpook Mathematical Journal
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    • v.61 no.3
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    • pp.487-494
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    • 2021
  • 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.

LOCALLY COMPLETE INTERSECTION IDEALS IN COHEN-MACAULAY LOCAL RINGS

  • Kim, Mee-Kyoung
    • Communications of the Korean Mathematical Society
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    • v.9 no.2
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    • pp.261-264
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    • 1994
  • Throughout this paper, all rings are assumed to be commutative with identity. By a local ring (R, m), we mean a Noetherian ring R which has the unique maximal ideal m. By dim(R) we always mean the Krull dimension of R. Let I be an ideal in a ring R and t an indeterminate over R. Then the Rees algebra R[It] is defined to be(omitted)

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