insight - ScientificComputing - # Transfer Krull Orders

A Characterization of Transfer Krull Orders in Dedekind Domains with Torsion Class Group

Q: Could there be alternative arithmetic invariants that differentiate transfer Krull orders from their encompassing Dedekind domains?

While transfer Krull orders inherit many arithmetic properties from their encompassing Dedekind domains, subtle differences might exist, detectable through alternative invariants. Here are some possibilities: 1. Refined Elasticity Measurements: Elasticity Function: The elasticity of an element measures the maximum ratio between lengths in different factorizations. While both the order and the Dedekind domain might have the same set of elasticities, the elasticity function, which considers the elasticity of each element, could differ. Local Elasticity: Investigating the elasticity within localizations at specific prime ideals, particularly those dividing the conductor, might reveal discrepancies not captured by the global elasticity. 2. Sets of Distances and Catenary Degree: Structure of Sets of Distances: The set of distances measures the gaps between consecutive lengths in factorizations. While Theorem 1 implies that the sets of lengths coincide, the structure and distribution of distances within these sets might differ. Catenary Degree: The catenary degree quantifies the complexity of factorizations by measuring the maximum length of chains needed to connect different factorizations of an element. Subtle differences in the catenary degree between the order and the Dedekind domain might exist. 3. Invariants Related to Prime Ideals: Distribution of Prime Ideals in Ideal Classes: While every class in the class group contains a prime ideal, the distribution and density of prime ideals within each class might differ between the order and the Dedekind domain. Behavior of Prime Ideals under Localization: Analyzing how prime ideals, particularly those dividing the conductor, behave under localization and how this affects factorization properties could reveal distinctions.

Q: What are the implications of this research for the study of factorization properties in other algebraic structures, such as rings of integer-valued polynomials?

The research on transfer Krull orders has implications for the study of factorization properties in other algebraic structures, particularly rings of integer-valued polynomials, by offering: 1. New Avenues for Investigation: Identifying Transfer Krull Subrings: The characterization of transfer Krull orders provides a blueprint for investigating whether subrings of rings of integer-valued polynomials can be transfer Krull. This opens up a new line of inquiry within the factorization theory of these rings. Exploring Connections with Value Sets: The techniques used in analyzing the localizations of transfer Krull orders could be adapted to study the factorization properties of integer-valued polynomials in relation to their value sets. 2. Tools for Analyzing Factorization: Transferring Arithmetic Information: If a subring of a ring of integer-valued polynomials is identified as transfer Krull, the transfer homomorphism becomes a powerful tool for transferring arithmetic information from a Krull monoid, whose arithmetic is well-understood, to the subring. Understanding Factorization Complexity: The study of invariants like elasticity, sets of distances, and catenary degree in the context of transfer Krull orders could provide insights into the factorization complexity of specific subrings of rings of integer-valued polynomials. 3. Potential for Generalizations: Abstracting Key Concepts: The concepts of conductor, regular elements, and the interplay between local and global factorization properties, central to the study of transfer Krull orders, could be abstracted and applied to other algebraic structures beyond rings of integer-valued polynomials. Developing New Characterizations: The techniques used to characterize transfer Krull orders might inspire the development of similar characterizations for other classes of rings and monoids, enriching the study of factorization theory in diverse algebraic settings.

Core Concepts

This research paper characterizes the specific conditions under which orders within Dedekind domains with torsion class groups can be classified as transfer Krull, highlighting the arithmetic implications of this classification.

Abstract

Bibliographic Information: Rago, B. (2024). A characterization of transfer Krull orders in Dedekind domains with torsion class group. arXiv preprint arXiv:2411.00271v1.
Research Objective: The paper aims to establish an algebraic characterization of transfer Krull orders within Dedekind domains possessing torsion class groups.
Methodology: The author utilizes concepts from factorization theory, focusing on transfer homomorphisms and their properties. The study delves into the arithmetic relationships between orders and their corresponding Dedekind domains, particularly examining the behavior of regular elements and localizations.
Key Findings: The research reveals that an order within a Dedekind domain with a torsion class group, excluding the case where the class group has order 2, is transfer Krull if and only if specific conditions related to its units and atom valuations are met. Notably, the inclusion map from the order to the Dedekind domain acts as a transfer homomorphism in such cases.
Main Conclusions: The study concludes that the arithmetic properties of a Dedekind domain with a torsion class group are largely mirrored in its transfer Krull orders. This finding is significant because it allows for the application of established arithmetic results for Dedekind domains to the less-studied realm of transfer Krull orders.
Significance: This research contributes significantly to the understanding of non-Krull domains, particularly orders within Dedekind domains. By characterizing transfer Krull orders, the study provides valuable insights into their arithmetic structure and facilitates further exploration of their properties.
Limitations and Future Research: The paper acknowledges the special case where the class group of the Dedekind domain has order 2, suggesting further investigation into the necessity of the additional assumption regarding the Picard group in this scenario.

Customize Summary

Rewrite with AI

Generate Citations

Translate Source

To Another Language

Generate MindMap

from source content

Visit Source

arxiv.org

Stats

Quotes

Key Insights Distilled From

A characterization of transfer Krull orders in Dedekind domains with torsion class group

by Balint Rago at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00271.pdf

A characterization of transfer Krull orders in Dedekind domains with torsion class group

Deeper Inquiries

How can the characterization of transfer Krull orders be extended to more general settings beyond Dedekind domains?

Extending the characterization of transfer Krull orders beyond Dedekind domains poses a significant challenge in factorization theory. Here are some potential avenues for exploration:
1. Weakening the Krull Property:

Moving to Weakly Krull Domains: One natural step is to consider weakly Krull domains, a broader class containing Dedekind domains. These domains share some features with Krull domains but allow for more complex factorization behavior.  The existing literature already explores transfer homomorphisms in the context of T-block monoids for weakly Krull domains. The challenge lies in refining these results to obtain a characterization similar to Theorem 1.
Exploring C-Monoids: C-monoids provide an abstract framework encompassing various arithmetical structures, including Krull and weakly Krull monoids. Investigating transfer Krull monoids within the framework of C-monoids could offer a more general perspective.
2. Relaxing Conditions on the Class Group:

Handling Non-Torsion Class Groups: Theorem 1 relies heavily on the torsion property of the class group.  Extending the results to Dedekind domains with non-torsion class groups would require new techniques.  The interplay between the infinite order elements in the class group and the factorization properties of the order would need careful analysis.
Considering Class Groups with Elements without Prime Ideals: The assumption that every class contains a prime ideal is crucial for the existence of the transfer homomorphism to the monoid of zero-sum sequences. Relaxing this condition would necessitate exploring alternative target monoids for the transfer homomorphism.
3. Investigating Specific Classes of Domains:

Focusing on Orders in Quadratic Number Fields:  Orders in quadratic number fields provide a concrete and well-studied setting for exploring generalizations.  Analyzing the obstructions to being transfer Krull in this specific case could offer insights for broader extensions.
Examining Rings of Integer-Valued Polynomials: Rings of integer-valued polynomials present another interesting class of domains where factorization theory plays a crucial role.  Investigating the conditions under which these rings, or their subrings, become transfer Krull could lead to new connections between factorization theory and polynomial rings.

Could there be alternative arithmetic invariants that differentiate transfer Krull orders from their encompassing Dedekind domains?

While transfer Krull orders inherit many arithmetic properties from their encompassing Dedekind domains, subtle differences might exist, detectable through alternative invariants. Here are some possibilities:
1. Refined Elasticity Measurements:

Elasticity Function: The elasticity of an element measures the maximum ratio between lengths in different factorizations.  While both the order and the Dedekind domain might have the same set of elasticities, the elasticity function, which considers the elasticity of each element, could differ.
Local Elasticity: Investigating the elasticity within localizations at specific prime ideals, particularly those dividing the conductor, might reveal discrepancies not captured by the global elasticity.
2. Sets of Distances and Catenary Degree:

Structure of Sets of Distances: The set of distances measures the gaps between consecutive lengths in factorizations. While Theorem 1 implies that the sets of lengths coincide, the structure and distribution of distances within these sets might differ.
Catenary Degree: The catenary degree quantifies the complexity of factorizations by measuring the maximum length of chains needed to connect different factorizations of an element.  Subtle differences in the catenary degree between the order and the Dedekind domain might exist.
3. Invariants Related to Prime Ideals:

Distribution of Prime Ideals in Ideal Classes: While every class in the class group contains a prime ideal, the distribution and density of prime ideals within each class might differ between the order and the Dedekind domain.
Behavior of Prime Ideals under Localization: Analyzing how prime ideals, particularly those dividing the conductor, behave under localization and how this affects factorization properties could reveal distinctions.

What are the implications of this research for the study of factorization properties in other algebraic structures, such as rings of integer-valued polynomials?

The research on transfer Krull orders has implications for the study of factorization properties in other algebraic structures, particularly rings of integer-valued polynomials, by offering:
1. New Avenues for Investigation:

Identifying Transfer Krull Subrings: The characterization of transfer Krull orders provides a blueprint for investigating whether subrings of rings of integer-valued polynomials can be transfer Krull. This opens up a new line of inquiry within the factorization theory of these rings.
Exploring Connections with Value Sets: The techniques used in analyzing the localizations of transfer Krull orders could be adapted to study the factorization properties of integer-valued polynomials in relation to their value sets.
2. Tools for Analyzing Factorization:

Transferring Arithmetic Information: If a subring of a ring of integer-valued polynomials is identified as transfer Krull, the transfer homomorphism becomes a powerful tool for transferring arithmetic information from a Krull monoid, whose arithmetic is well-understood, to the subring.
Understanding Factorization Complexity: The study of invariants like elasticity, sets of distances, and catenary degree in the context of transfer Krull orders could provide insights into the factorization complexity of specific subrings of rings of integer-valued polynomials.
3. Potential for Generalizations:

Abstracting Key Concepts: The concepts of conductor, regular elements, and the interplay between local and global factorization properties, central to the study of transfer Krull orders, could be abstracted and applied to other algebraic structures beyond rings of integer-valued polynomials.
Developing New Characterizations: The techniques used to characterize transfer Krull orders might inspire the development of similar characterizations for other classes of rings and monoids, enriching the study of factorization theory in diverse algebraic settings.