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An integrally closed ring which is not the intersection of valuation rings


Author: Joachim Gräter
Journal: Proc. Amer. Math. Soc. 107 (1989), 333-336
MSC: Primary 13B20; Secondary 13A18
DOI: https://doi.org/10.1090/S0002-9939-1989-0972231-9
MathSciNet review: 972231
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Abstract: Each commutative ring $ R$ which is integrally closed in its total quotient ring $ T(R)$ is the intersection of all paravaluation rings of $ T(R)$ containing $ R$. In this note an example is given that shows that this statement is not true with "valuation rings" instead of "paravaluation rings". This is an answer of a question asked by J. A. Huckaba in [3].


References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Elements of mathematics, commutative algebra, Hermann, Paris, Addison-Wesley Publishing Co., Reading, Massachusetts, 1972 MR 0360549 (50:12997)
  • [2] J. Gräter, Integral closure and valuation rings with zero-divisors, Studia Sci. Math. Hungarica 17 (1982), 457-458 MR 761562 (85h:13005)
  • [3] J. A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, M. Dekker, Inc., New York and Basel, 1988 MR 938741 (89e:13001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0972231-9
Keywords: Commutative rings, integral closure, valuation rings
Article copyright: © Copyright 1989 American Mathematical Society

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