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Almost Dedekind domains without radical factorization

  • Dario Spirito ORCID logo EMAIL logo
Published/Copyright: January 30, 2023
Forum Mathematicum
From the journal Forum Mathematicum Volume 35 Issue 2

Abstract

We study almost Dedekind domains with respect to the failure of ideals to have radical factorization, that is, we study how to measure how far an almost Dedekind domain is from being an SP-domain. To do so, we consider the maximal space ℳ = Max ⁡ ( R ) of an almost Dedekind domain R, interpreting its (fractional) ideals as maps from ℳ to ℤ , and looking at the continuity of these maps when ℳ is endowed with the inverse topology and ℤ with the discrete topology. We generalize the concept of critical ideals by introducing a well-ordered chain of closed subsets of ℳ (of which the set of critical ideals is the first step) and use it to define the class of SP-scattered domains, which includes the almost Dedekind domains such that ℳ is scattered and, in particular, the almost Dedekind domains such that ℳ is countable. We show that for this class of rings the group Inv ⁢ ( R ) is free by expressing it as a direct sum of groups of continuous maps, and that, for every length function ℓ on R and every ideal I of R, the length of R / I is equal to the length of R / rad ⁡ ( I ) .

Keywords: Almost Dedekind domains; critical maximal ideals; free abelian group; invertible ideals; divisorial ideals; extremally disconnected spaces; length function
MSC 2010: 13F05; 13A15; 06F15

Communicated by Manfred Droste

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Received: 2022-01-24
Revised: 2022-11-10
Published Online: 2023-01-30
Published in Print: 2023-03-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the asymptotics of the shifted sums of Hecke eigenvalue squares
  3. On the splitting conjecture in the hybrid model for the Riemann zeta function
  4. Almost Dedekind domains without radical factorization
  5. Groups in which the centralizer of any non-central primary element is maximal
  6. GKM actions on cohomogeneity one manifolds
  7. Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
  8. Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
  9. Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
  10. Measures of noncompactness of interpolated polynomials
  11. The homotopy category of acyclic complexes of pure-projective modules
  12. The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
  13. Sharp Li–Yau inequalities for Dunkl harmonic oscillators
  14. A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
  15. Free polynilpotent groups and the Magnus property
https://doi.org/10.1515/forum-2022-0033
Keywords for this article
Almost Dedekind domains; critical maximal ideals; free abelian group; invertible ideals; divisorial ideals; extremally disconnected spaces; length function

Articles in the same Issue

  1. Frontmatter
  2. On the asymptotics of the shifted sums of Hecke eigenvalue squares
  3. On the splitting conjecture in the hybrid model for the Riemann zeta function
  4. Almost Dedekind domains without radical factorization
  5. Groups in which the centralizer of any non-central primary element is maximal
  6. GKM actions on cohomogeneity one manifolds
  7. Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
  8. Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
  9. Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
  10. Measures of noncompactness of interpolated polynomials
  11. The homotopy category of acyclic complexes of pure-projective modules
  12. The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
  13. Sharp Li–Yau inequalities for Dunkl harmonic oscillators
  14. A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
  15. Free polynilpotent groups and the Magnus property
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