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A characterization of large Dedekind domains

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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

  1. Recall that a discrete valuation ring is a principal ideal domain V with a unique nonzero prime ideal \({\mathfrak {m}}\).

  2. Here, S or T could be trivial.

  3. Observe that \(X\in V\) is not invertible in V, and so V is not a field.

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Acknowledgements

The author thanks the anonymous referee for suggesting the second proof of Theorem 2 as well as Proposition 3.

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Correspondence to Greg Oman.

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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

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