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PreprintCC BY 4.0arXiv

Class groups of polynomial Krull domains

  • Djamila Ait Elhadi
  • Giulio Peruginelli
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Abstract

A polynomial Krull domain is a Krull domain contained between ℤ[X] and ℚ[X]. The domains of this class are precisely the Noetherian integrally closed domains between these polynomial rings by a result of Heinzer, and they have been recently classified by Chang and the second author of this paper as rings of polynomials which are integer-valued over a finite set of ultrametric balls in ℂp, as p ranges through the set of primes in ℤ. We show here that every ideal class of a polynomial Krull domain contains a prime ideal. Furthermore, given such a Krull domain R, we prove that there exists a polynomial Krull domain R' that is Dedekind, contains R and has the same class group as R.

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Cite

BibTeXRIS
BibTeX
@misc{elhadi2026class,
  title = {{Class groups of polynomial Krull domains}},
  author = {Djamila Ait Elhadi and Giulio Peruginelli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38457},
  eprint = {2610.38457},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38457},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Class groups of polynomial Krull domains
AU  - Djamila Ait Elhadi
AU  - Giulio Peruginelli
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38457
UR  - https://arxiv.org/abs/2610.38457
AB  - A polynomial Krull domain is a Krull domain contained between ℤ[X] and ℚ[X]. The domains of this class are precisely the Noetherian integrally closed domains between these polynomial rings by a result of Heinzer, and they have been recently classified by Chang and the second author of this paper as rings of polynomials which are integer-valued over a finite set of ultrametric balls in ℂ_p, as p ranges through the set of primes in ℤ. We show here that every ideal class of a polynomial Krull domain contains a prime ideal. Furthermore, given such a Krull domain R, we prove that there exists a polynomial Krull domain R' that is Dedekind, contains R and has the same class group as R.
ER  -