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A compactness property for prime ideals in Noetherian rings


Authors: Clive M. Reis and T. M. Viswanathan
Journal: Proc. Amer. Math. Soc. 25 (1970), 353-356
MSC: Primary 13.50
DOI: https://doi.org/10.1090/S0002-9939-1970-0254031-6
MathSciNet review: 0254031
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Abstract: A ring $ R$ is compactly packed by prime ideals if whenever an ideal $ I$ of $ R$ is contained in the union of a family of prime ideals of $ R,I$ is actually contained in one of the prime ideals of the family. It is shown that a commutative Noetherian ring is compactly packed if and only if every prime ideal is the radical of a principal ideal. For Dedekind domains this is equivalent to the torsion of the ideal class group and again to the existence of distinguished elements for the essential valuations. If a Noetherian ring $ R$ is compactly packed then Krull dim. $ R \leqq 1$. Also a Krull domain $ R$ is compactly packed if and only if it is a Dedekind domain with torsion ideal class group.


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  • [1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • [2] N. Bourbaki, Éléments de mathématique. Fasc. XXXI. Algèbre commutative. Chapitre 7: Diviseurs, Actualités Scientifiques et Industrielles, No. 1314, Hermann, Paris, 1965 (French). MR 0260715
  • [3] Luther Claborn, Every abelian group is a class group, Pacific J. Math. 18 (1966), 219–222. MR 0195889
  • [4] Luther Claborn, Dedekind domains: Overrings and semi-prime elements, Pacific J. Math. 15 (1965), 799–804. MR 0188243
  • [5] Luther Claborn, Dedekind domains and rings of quotients, Pacific J. Math. 15 (1965), 59–64. MR 0178005
  • [6] Harry Pollard, The Theory of Algebraic Numbers, Carus Monograph Series, no. 9, The Mathematical Association of America, Buffalo, N. Y., 1950. MR 0037319
  • [7] P. Ribenboim, Theory of valuations, Lecture Notes, Queen's University, Kingston, 1967.
  • [8] Pierre Samuel, Progrès récents d’algèbre locale, Rédaction par Renzo Piccinini et Paulo Ribenboim. Notas de Matemática, No. 19, Instituto de Matemática Pura e Aplicada do Conselho Nacional de Pesquisas, Rio de Janeiro, 1959 (Italian). MR 0144918
  • [9] Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
  • [10] Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR 0120249

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0254031-6
Keywords: Noetherian ring, Dedekind domain, Krull domain, Krull dimension
Article copyright: © Copyright 1970 American Mathematical Society