Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NOTE ON w-GD DOMAINS

  • Zhou, Dechuan (School of Science Southwest University of Science and Technology)
  • Received : 2019.11.16
  • Accepted : 2020.08.21
  • Published : 2020.11.30
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Abstract

Let S and T be w-linked extension domains of a domain R with S ⊆ T. In this paper, we define what satisfying the wR-GD property for S ⊆ T means and what being wR- or w-GD domains for T means. Then some sufficient conditions are given for the wR-GD property and wR-GD domains. For example, if T is wR-integral over S and S is integrally closed, then the wR-GD property holds. It is also given that S is a wR-GD domain if and only if S ⊆ T satisfies the wR-GD property for each wR-linked valuation overring T of S, if and only if S ⊆ (S[u])w satisfies the wR-GD property for each element u in the quotient field of S, if and only if S𝔪 is a GD domain for each maximal wR-ideal 𝔪 of S. Then we focus on discussing the relationship among GD domains, w-GD domains, wR-GD domains, Prüfer domains, PνMDs and PwRMDs, and also provide some relevant counterexamples. As an application, we give a new characterization of PwRMDs. We show that S is a PwRMD if and only if S is a wR-GD domain and every wR-linked overring of S that satisfies the wR-GD property is wR-flat over S. Furthermore, examples are provided to show these two conditions are necessary for PwRMDs.

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