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A note on intersections of valuation ideals


Author: Charles H. Brase
Journal: Proc. Amer. Math. Soc. 38 (1973), 37-39
DOI: https://doi.org/10.1090/S0002-9939-1973-0309915-X
MathSciNet review: 0309915
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Abstract | References | Additional Information

Abstract: In this note it is proved that if $ R$ is an integral domain the set of valuation ideals of $ R$ is closed under intersection if and only if the integral closure of $ R$ is a valuation ring. Let $ S$ be a domain which is integrally dependent on $ R$ and contains the integral closure of $ R$. Then the set of valuation ideals of $ R$ is the same as the set of ideals of $ R$ which contract from ideals of $ S$ if and only if the set of valuation ideals of $ R$ is closed under intersection.


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

  • [1] R. W. Gilmer, Jr., Contracted ideals with respect to integral extensions, Duke Math. J. 34 (1967), 561-571. MR 36 #1428. MR 0218341 (36:1428)
  • [2] -, Multiplicative ideal theory, Queen's Papers in Pure and Appl. Math., no. 12, Queen's University, Kingston, Ont., 1968. MR 37 #5198. MR 0229624 (37:5198)
  • [3] R. Gilmer and J. Ohm, Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 117 (1965), 237-250. MR 30 #114. MR 0169871 (30:114)
  • [4] M. Larsen and P. McCarthy, Multiplicative theory of ideals, Academic Press, New York, 1971. MR 0414528 (54:2629)
  • [5] O. Zariski and P. Samuel, Commutative algebra. Vol. 1, University Ser. in Higher Math., Van Nostrand, Princeton, N.J., 1958. MR 19, 833. MR 0090581 (19:833e)
  • [6] -, Commutative algebra. Vol. 2, University Ser. in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #11006.


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0309915-X
Keywords: Valuation ideal, completion, integral closure
Article copyright: © Copyright 1973 American Mathematical Society

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