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On the arithmetic of stable domains
Aqsa Bashir1, Alfred Geroldinger1, Andreas Reinhart1
1Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, NAWI Graz, Graz, Austria.
Communications in Algebra
|September 3, 2021
Summary
Stable rings are defined by their projective ideals. This study explores the arithmetic properties of stable integral domains and their ideal semigroups, particularly within stable orders in Dedekind domains.
Area of Science:
- Commutative Algebra
- Ring Theory
- Algebraic Number Theory
Background:
- Introduces the concept of stable rings, where non-zero ideals are projective over their endomorphism rings.
- Highlights the historical significance and extensive research on stable rings since the 1960s.
- Mentions known relationships between stability, divisoriality, and the 2-generator property.
Purpose of the Study:
- To investigate the arithmetic properties of stable integral domains.
- To focus on the arithmetic of semigroups of ideals within stable orders in Dedekind domains.
Main Methods:
- Analysis of ideal structures in commutative rings.
- Exploration of endomorphism rings and projectivity.
- Study of semigroups of ideals in specific algebraic structures.
Main Results:
- Characterizes the arithmetic of stable integral domains.
- Provides insights into the structure of ideal semigroups in stable orders.
- Connects stability properties to arithmetic behaviors in Dedekind domain settings.
Conclusions:
- Advances the understanding of stable rings by focusing on their arithmetic.
- Offers new perspectives on ideal semigroups in the context of stable orders.
- Contributes to the broader study of algebraic number theory and commutative algebra.
Keywords:
13A0513A1513F0513H10Catenary degreesMori domainsfactorizationssets of lengthsstable domainsMore Related Videos
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