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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Noetherian domains with many more elements than height-one primes


Author: D. D. Anderson
Journal: Proc. Amer. Math. Soc. 123 (1995), 2971-2974
MSC: Primary 13F05; Secondary 13F07
MathSciNet review: 1264799
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Abstract: Extending previous results of L. Claborn and H.W. Lenstra, Jr., we show that if D is a Krull domain with a set of height-one primes $ {X^{(1)}}$ that satisfies either (1) D contains a subset k with $ \vert k\vert > \vert{X^{(1)}}\vert$ and for $ \mu \ne \lambda \in k,\mu - \lambda $ is a unit, or (2) $ \vert D\vert > \vert{X^{(1)}}{\vert^{{\aleph _0}}}$, then D is a Euclidean domain. We also show that any Noetherian ring satisfying (1) or Noetherian domain satisfying (2) has Krull dimension at most one.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1264799-5
PII: S 0002-9939(1995)1264799-5
Article copyright: © Copyright 1995 American Mathematical Society