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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An integrally closed ring which is not the intersection of valuation rings
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by Joachim Gräter PDF
Proc. Amer. Math. Soc. 107 (1989), 333-336 Request permission

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
  • 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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Additional Information
  • © Copyright 1989 American Mathematical Society
  • 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