Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EVERY ABELIAN GROUP IS THE CLASS GROUP OF A RING OF KRULL TYPE

  • Chang, Gyu Whan (Department of Mathematics Education Incheon National University)
  • Received : 2020.01.08
  • Accepted : 2020.06.25
  • Published : 2021.01.01
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Abstract

Let Cl(A) denote the class group of an arbitrary integral domain A introduced by Bouvier in 1982. Then Cl(A) is the ideal class (resp., divisor class) group of A if A is a Dedekind or a Prüfer (resp., Krull) domain. Let G be an abelian group. In this paper, we show that there is a ring of Krull type D such that Cl(D) = G but D is not a Krull domain. We then use this ring to construct a Prüfer ring of Krull type E such that Cl(E) = G but E is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.

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Acknowledgement

The author would like to thank the anonymous referee for his/her helpful comments and suggestions which improved the original version of this paper greatly. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B06029867).

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