Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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A Note on S-Noetherian Domains

  • LIM, JUNG WOOK (Department of Mathematics, Kyungpook National University)
  • Received : 2014.09.10
  • Accepted : 2014.11.07
  • Published : 2015.09.23
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Abstract

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.

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