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A local characterization of Noetherian and Dedekind rings


Author: Yves Lequain
Journal: Proc. Amer. Math. Soc. 94 (1985), 369-370
MSC: Primary 13E05; Secondary 13F05
DOI: https://doi.org/10.1090/S0002-9939-1985-0787874-1
MathSciNet review: 787874
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Abstract: Let $ R$ be a ring and $ M$ a maximal ideal of $ R$. Then $ R$ is Noetherian if and only if every ideal contained in $ M$ is finitely generated; $ R$ is Dedekind if and only if every nonzero ideal contained in $ M$ is invertible.


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

  • [1] I. S. Cohen, commutative rings with restricted minimum condition, Duke Math. J. 17 (1950), 27-42. MR 0033276 (11:413g)
  • [2] J. Ohm and D. E. Rush, The finiteness of $ I$ when $ R[X]/I$ is flat, Trans. Amer. Math. Soc. 171 (1972), 377-408. MR 0306176 (46:5303)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0787874-1
Article copyright: © Copyright 1985 American Mathematical Society

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