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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?)

  • Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
  • J. Gräter, Integral closure and valuation rings with zero-divisors, Studia Sci. Math. Hungar. 17 (1982), no. 1-4, 457–458. MR 761562
  • 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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Keywords: Commutative rings, integral closure, valuation rings
Article copyright: © Copyright 1989 American Mathematical Society