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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, Reading, Mass., 1969. MR 0242802 (39:4129)
  • [2] N. Bourbaki, Algèbre commutative. Chapitre 7: Diviseurs, Hermann, Paris, 1965. MR 0260715 (41:5339)
  • [3] L. Claborn, Every abelian group is a class group, Pacific J. Math. 18 (1966), 219-222. MR 33 #4085. MR 0195889 (33:4085)
  • [4] -, Dedekind domains: Overrings and semi-prime elements, Pacific J. Math. 15 (1965), 799-804. MR 32 #5682. MR 0188243 (32:5682)
  • [5] -, Dedekind domains and rings of quotients, Pacific J. Math. 15 (1965), 59-64. MR 31 #2263. MR 0178005 (31:2263)
  • [6] H. Pollard, The theory of algebraic numbers, Carus Monograph Series no. 9, The Mathematical Association of America, Buffalo, New York, 1950. MR 12, 243. MR 0037319 (12:243g)
  • [7] P. Ribenboim, Theory of valuations, Lecture Notes, Queen's University, Kingston, 1967.
  • [8] P. Samuel, Progrès recénts d'algèbre locale, Notas Math., no. 19, Instituto de Matemática Pura e Aplicada de Conselho Nacionale de Pesquisas, Rio de Janeiro, 1959. MR 26 #2458. MR 0144918 (26:2458)
  • [9] O. Zariski and P. Samuel, Commutative algebra. Vol. I, The University Series in Higher Mathematics, Van Nostrand, Princeton, N. J., 1958. MR 19, 833. MR 0090581 (19:833e)
  • [10] -, Commutative algebra. Vol. II, The University Series in Higher Mathematics, Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)

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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

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