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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
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] Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
  • [2] J. Gräter, Integral closure and valuation rings with zero-divisors, Studia Sci. Math. Hungar. 17 (1982), no. 1-4, 457–458. MR 761562
  • [3] James A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, vol. 117, Marcel Dekker, Inc., New York, 1988. MR 938741

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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