A characterization of transfer Krull orders in Dedekind domains with torsion class group
Pith reviewed 2026-05-23 18:37 UTC · model grok-4.3
The pith
Orders in Dedekind domains with torsion class group are characterized by admitting a transfer homomorphism to the monoid of zero-sum sequences.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We establish a characterization (under some natural conditions) of those orders in Dedekind domains which allow a transfer homomorphism to a monoid of zero-sum sequences. As a consequence, the inclusion map to the Dedekind domain is a transfer homomorphism, with the exception of a particular case.
What carries the argument
Transfer homomorphism to the monoid of zero-sum sequences, which preserves arithmetic properties between the order and the Dedekind domain.
If this is right
- The inclusion map from the order to the Dedekind domain is a transfer homomorphism except in one particular case.
- Orders satisfying the characterization share the arithmetic properties of the Dedekind domain.
- The arithmetic of such orders is as well understood as the arithmetic of Krull domains.
Where Pith is reading between the lines
- The result may allow direct transfer of known factorization theorems from Dedekind domains to these special orders.
- It could help classify when non-Dedekind rings still exhibit Krull-like arithmetic.
Load-bearing premise
The Dedekind domain has a torsion class group and the orders satisfy unspecified natural conditions.
What would settle it
An explicit order inside a Dedekind domain with torsion class group that admits a transfer homomorphism to the zero-sum sequence monoid but fails the stated characterization, or that satisfies the characterization but does not admit such a homomorphism.
read the original abstract
We establish a characterization (under some natural conditions) of those orders in Dedekind domains which allow a transfer homomorphism to a monoid of zero-sum sequences. As a consequence, the inclusion map to the Dedekind domain is a transfer homomorphism, with the exception of a particular case. The arithmetic of Krull and Dedekind domains is well understood, and the existence of a transfer homomorphism implies that the order and the associated Dedekind domain share the same arithmetic properties. This is not the case for arbitrary orders in Dedekind domains.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript establishes a characterization (under some natural conditions) of those orders in Dedekind domains with torsion class group which admit a transfer homomorphism to the monoid of zero-sum sequences over the class group. As a consequence, the inclusion map into the Dedekind domain is itself a transfer homomorphism, except in one identified case. The arithmetic of Krull and Dedekind domains is well understood, and the existence of such a transfer homomorphism implies that the order and the Dedekind domain share the same arithmetic properties (unlike arbitrary orders).
Significance. If the result holds, it provides a concrete criterion for when orders in Dedekind domains inherit the full arithmetic structure (e.g., factorization properties) of the ambient domain via transfer homomorphisms, extending the theory of Krull monoids in a useful way. The argument relies on established tools from factorization theory rather than ad-hoc constructions, which strengthens the result.
minor comments (1)
- [Abstract / §1] The abstract and introduction refer to 'some natural conditions' without a brief enumeration or forward reference to the precise statement in the main theorem; adding this would improve readability for readers outside the immediate subfield.
Simulated Author's Rebuttal
We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript. No major comments were raised in the report.
Circularity Check
No significant circularity
full rationale
The paper states a characterization theorem for orders in Dedekind domains with torsion class group that admit transfer homomorphisms to zero-sum sequence monoids, relying on the standard framework of Krull monoids and factorization theory. The torsion class group is an explicit hypothesis of the setting rather than a derived quantity, and the consequence about the inclusion map follows from the characterization under the stated natural conditions. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described structure; the derivation chain rests on externally established tools in the field without reducing the central claim to its own inputs by construction.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The class group of the Dedekind domain is torsion
- ad hoc to paper Natural conditions on the orders
Reference graph
Works this paper leans on
-
[1]
N.R. Baeth and D. Smertnig,, Lattices over Bass rings and graph agglomerations , Algebras and Representa- tion Theory 25 (2021), 669 – 704
work page 2021
-
[2]
A. Bashir and A. Reinhart, On transfer Krull monoids , Semigroup Forum 105 (2022), 73 – 95
work page 2022
-
[3]
J.P. Bell, K. Brown, Z. Nazemian, and D. Smertnig, On noncommutative bounded factorization domains and prime rings , J. Algebra 622 (2023), 404 – 449
work page 2023
-
[4]
J. Brantner, A. Geroldinger, and A. Reinhart, On monoids of ideals of orders in quadratic number fields , in Advances in Rings, Modules, and Factorizations, vol. 321, S pringer, 2020, pp. 11 – 54
work page 2020
-
[5]
G.W. Chang and A. Geroldinger, On Dedekind domains whose class groups are direct sums of cyc lic groups , J. Pure Appl. Algebra 228 (2024), Paper No. 107470, 14pp. 14 BALINT RAGO
work page 2024
- [6]
- [7]
-
[8]
W. Gao, C. Liu, S. Tringali, and Q. Zhong, On half-factoriality of transfer Krull monoids , Commun. Algebra 49 (2021), 409 – 420
work page 2021
-
[9]
A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic The- ory, Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC , 2006
work page 2006
-
[10]
A. Geroldinger, F. Kainrath, and A. Reinhart, Arithmetic of seminormal weakly Krull monoids and domains , J. Algebra 444 (2015), 201 – 245
work page 2015
-
[11]
A. Geroldinger and I. Ruzsa, Combinatorial Number Theory and Additive Group Theory , Advanced Courses in Mathematics - CRM Barcelona, Birkh¨ auser, 2009
work page 2009
-
[12]
A. Geroldinger, W.A. Schmid, and Q. Zhong, Systems of sets of lengths: transfer Krull monoids versus we akly Krull monoids , in Rings, Polynomials, and Modules, Springer, Cham, 2017, pp. 191 – 235
work page 2017
-
[13]
A. Geroldinger and Q. Zhong, Factorization theory in commutative monoids , Semigroup Forum 100 (2020), 22 – 51
work page 2020
-
[14]
D.J. Grynkiewicz, The Characterization of Finite Elasticities: Factorizati on Theory in Krull Monoids via Convex Geometry, Lecture Notes in Math., vol. 2316, Springer, 2022
work page 2022
-
[15]
F. Halter-Koch, An Invitation to Algebraic Numbers and Algebraic Functions , CRC Press, Boca Raton, FL, 2020
work page 2020
-
[16]
, Class Field Theory and L-Functions, CRC Press, Boca Raton, FL, 2022
work page 2022
-
[17]
F. Kainrath, On local half-factorial orders , in Arithmetic Properties of Commutative Rings and Monoids , Lect. Notes Pure Appl. Math., vol. 241, Chapman & Hall/CRC, 2 005, pp. 316 – 324
-
[18]
, On some arithmetical properties of noetherian domains , in Advances in Rings, Modules and Factor- izations, Springer Proc. Math. Stat., vol. 321, Springer, 2 020, pp. 217 – 222
-
[19]
Philipp, A precise result on the arithmetic of non-principal orders i n algebraic number fields , J
A. Philipp, A precise result on the arithmetic of non-principal orders i n algebraic number fields , J. Algebra Appl. 11, 1250087, 42pp
-
[20]
M. Picavet-L’Hermitte, Factorization in some orders with a PID as integral closure , Algebraic Number Theory and Diophantine Analysis (F. Halter-Koch and R. Tich y, eds.), Walter de Gruyter, 2000, pp. 365 – 390
work page 2000
-
[21]
Pollack, Half-factorial real quadratic orders , Arch
P. Pollack, Half-factorial real quadratic orders , Arch. Math. (Basel) 122 (2024), 491 – 500
work page 2024
-
[22]
Number Theory 267 (2025), 80 – 100
, Maximally elastic quadratic fields , J. Number Theory 267 (2025), 80 – 100
work page 2025
-
[23]
B. Rago, A characterization of half-factorial orders in algebraic n umber fields, Acta Arith., to appear (2024)
work page 2024
-
[24]
Reinhart, On orders in quadratic number fields with unusual sets of dist ances, Acta Arith
A. Reinhart, On orders in quadratic number fields with unusual sets of dist ances, Acta Arith. 211 (2023), 61 – 92
work page 2023
-
[25]
W.A. Schmid, Some recent results and open problems on sets of lengths of Kr ull monoids with finite class group, in Multiplicative Ideal Theory and Factorization Theory, Springer, 2016, pp. 323 – 352
work page 2016
-
[26]
Smertnig, Sets of lengths in maximal orders in central simple algebras , J
D. Smertnig, Sets of lengths in maximal orders in central simple algebras , J. Algebra 390 (2013), 1 – 43
work page 2013
-
[27]
, Factorizations in bounded hereditary noetherian prime rin gs, Proc. Edinburgh Math. Soc. 62 (2019), 395 – 442. University of Graz, NA WI Graz, Department of Mathematics an d Scientific Computing, Hein- richstraße 36, 8010 Graz, Austria Email address : balint.rago@uni-graz.at
work page 2019
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.