Abstract
Let D be a commutative domain with identity, and let \({\mathcal {L}}(D)\) be the lattice of nonzero ideals of D. Say that D is ideal upper finite provided \({\mathcal {L}}(D)\) is upper finite, that is, every nonzero ideal of D is contained in but finitely many ideals of D. Now let \(\kappa >2^{\aleph _0}\) be a cardinal. We show that a domain D of cardinality \(\kappa \) is ideal upper finite if and only if D is a Dedekind domain. We also show (in ZFC) that this result is sharp in the sense that if \(\kappa \) is a cardinal such that \(\aleph _0\le \kappa \le 2^{\aleph _0}\), then there is an ideal upper finite domain of cardinality \(\kappa \) which is not Dedekind.
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Notes
Recall that a discrete valuation ring is a principal ideal domain V with a unique nonzero prime ideal \({\mathfrak {m}}\).
Here, S or T could be trivial.
Observe that \(X\in V\) is not invertible in V, and so V is not a field.
References
Chew, K., Lawn, S.: Residually finite rings. Can. J. Math. 22, 92–101 (1970)
Gilmer, R.: Multiplicative Ideal Theory. Corrected Reprint of the 1972 Edition. Queen’s Papers in Pure and Applied Mathematics, vol. 90. Queen’s University, Kingston, ON (1992)
Lang, S.: Algebra. Graduate Texts in Mathematics, Revised 3rd edn., vol. 211. Springer, New York (2002)
Mclean, K.R.: Commutative Artinian principal ideal rings. Proc. Lond. Math. Soc. 26(3), 249–272 (1973)
Oman, G.: Jónsson modules over Noetherian rings. Comm. Algebra 38(9), 3489–3498 (2010)
Oman, G.: Small and large ideals of an associative ring. J. Algebra Appl. 13(5), 1350151 (2014)
Zwann, M.B.J.: On commutative rings with only finitely many ideals. Thesis, Mathematisch Instituut, Universiteit Leiden. https://www.math.leidenuniv.nl/scripties/ZwaanBach.pdf (2014)
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Oman, G. A characterization of large Dedekind domains. Arch. Math. 115, 159–168 (2020). https://doi.org/10.1007/s00013-020-01443-6
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DOI: https://doi.org/10.1007/s00013-020-01443-6
Keywords
- Dedekind domain
- Field of fractions
- Krull’s intersection theorem
- Principal ideal ring
- (Discrete) valuation (over) ring