Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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A Characterization of Dedekind Domains and ZPI-rings

  • Rostami, Esmaeil (Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman)
  • Received : 2017.02.03
  • Accepted : 2017.09.09
  • Published : 2017.10.23
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Abstract

It is well known that an integral domain D is a Dedekind domain if and only if D is a Noetherian almost Dedekind domain. In this paper, we show that an integral domain D is a Dedekind domain if and only if D is an almost Dedekind domain such that Max(D) is a Noetherian topological space as a subspace of Spec(D) with respect to the Zariski topology. We also give a new characterization of ZPI-rings.

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References

  1. F. W. Anderson and K .R. Fuller, Rings and categories of modules, Second edition, Graduate Texts in Mathematics 13, Springer-Verlag, Berlin-Heidelberg-New York, 1992.
  2. R. Gilmer, Rings in which semi-primary ideals are primary, Pacifie J. Math., 12(1962), 1273-1276 https://doi.org/10.2140/pjm.1962.12.1273
  3. R. Gilmer, Extension of results concerning rings in which semi-primary ideals, Duke Math. J., 31(1964), 73-78. https://doi.org/10.1215/S0012-7094-64-03106-0
  4. R. Gilmer, Multiplicative ideal theory, Pure and Applied Mathematics 12, Marcel Dekker, New York, 1972.
  5. M. Larsen, and P. McCarthy, Multiplicative theory of ideals, Pure and Applied Mathematics 43, Academic Press, New York, 1971.
  6. A. Loper, Almost Dedekind domains which are not Dedekind, in Multiplicative ideal theory in commutative algebra. A tribute to the work of Robert Gilmer (edited by W. Brewer, S. Glaz, W. Heinzer, and B. Olberding ) 276-292, Springer, New York, 2006.
  7. J. Ohm and R. L. Pendleton, Ring with Noetherian spectrum, Duke Math. J., 35(1968), 631-639. https://doi.org/10.1215/S0012-7094-68-03565-5