A note on intersections of valuation ideals
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- by Charles H. Brase PDF
- Proc. Amer. Math. Soc. 38 (1973), 37-39 Request permission
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
- Robert W. Gilmer Jr., Contracted ideals with respect to integral extensions, Duke Math. J. 34 (1967), 561–571. MR 218341
- Robert W. Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR 0229624
- Robert Gilmer and Jack Ohm, Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 117 (1965), 237–250. MR 169871, DOI 10.1090/S0002-9947-1965-0169871-4
- Max. D. Larsen and Paul J. McCarthy, Multiplicative theory of ideals, Pure and Applied Mathematics, Vol. 43, Academic Press, New York-London, 1971. MR 0414528
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581 —, Commutative algebra. Vol. 2, University Ser. in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #11006.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 38 (1973), 37-39
- DOI: https://doi.org/10.1090/S0002-9939-1973-0309915-X
- MathSciNet review: 0309915