scholarly journals Radicals of principal ideals and the class group of a Dedekind domain

2021 ◽  
Vol 314 (1) ◽  
pp. 219-231
Author(s):  
Dario Spirito
Keyword(s):  
Class Group ◽  
Dedekind Domain ◽  
Principal Ideals

scholarly journals The distribution of prime ideals of a Dedekind domain

1974 ◽  
Vol 11 (3) ◽  
pp. 429-441 ◽  
Author(s):  
Anne P. Grams
Keyword(s):  
Class Group ◽  
Sufficient Conditions ◽  
Torsion Group ◽  
Dedekind Domain ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Class Groups ◽  
Maximal Ideals ◽  
The Family ◽  
Necessary And Sufficient

Let G be an abelian group, and let S be a subset of G. Necessary and sufficient conditions on G and S are given in order that there should exist a Dedekind domain D with class group G with the property that S is the set of classes that contain maximal ideals of D. If G is a torsion group, then S is the set of classes containing the maximal ideals of D if and only if S generates G. These results are used to determine necessary and sufficient conditions on a family {Hλ} of subgroups of G in order that there should exist a Dedekind domain D with class group G such that {G/Hλ} is the family of class groups of the set of overrings of D. Several applications are given.

scholarly journals A Dedekind Domain with Nontrivial Class Group

2018 ◽  
Vol 125 (4) ◽  
pp. 356-359
Author(s):  
Vaibhav Pandey ◽  
Sagar Shrivastava ◽  
Balasubramanian Sury
Keyword(s):  
Class Group ◽  
Dedekind Domain

A QUANTITATIVE ASPECT OF NON-UNIQUE FACTORIZATIONS: THE NARKIEWICZ CONSTANTS

2011 ◽  
Vol 07 (06) ◽  
pp. 1463-1502 ◽  
Author(s):  
WEIDONG GAO ◽  
ALFRED GEROLDINGER ◽  
QINGHONG WANG
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Class Group ◽  
Algebraic Number Field ◽  
Quantitative Aspect ◽  
Ring Of Integers ◽  
New Methods ◽  
Principal Ideals ◽  
Combinatorial Number Theory ◽  
Irreducible Elements

Let K be an algebraic number field with non-trivial class group G and let [Formula: see text] be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let Fk (x) denote the number of non-zero principal ideals [Formula: see text] with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that Fk (x) behaves, for x → ∞, asymptotically like x( log x)-1+1/|G|( log log x) N k(G). We study N k (G) with new methods from Combinatorial Number Theory.

scholarly journals Almost-Dedekind rings

1994 ◽  
Vol 36 (1) ◽  
pp. 131-134 ◽  
Author(s):  
E. W. Johnson
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
Classical Result ◽  
Dedekind Domain ◽  
Ring Theory ◽  
Prime Ideals ◽  
Principal Ideals ◽  
Commutative Ring Theory ◽  
Lattice Of Ideals

Throughout we assume all rings are commutative with identity. We denote the lattice of ideals of a ringRbyL(R), and we denote byL(R)* the subposetL(R)−R.A classical result of commutative ring theory is the characterization of a Dedekind domain as an integral domainRin which every element ofL(R)* is a product of prime ideals (see Mori [5] for a history). This result has been generalized in a number of ways. In particular, rings which are not necessarily domains but which otherwise satisfy the hypotheses (i.e. general ZPI-rings) have been widely studied (see, for example, Gilmer [3]), as have rings in which only the principal ideals are assumed to satisfy the hypothesis (i.e. π-rings).

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Electoral Integrity in America

2018 ◽  
Keyword(s):  
News Media ◽  
Class Group ◽  
Public Trust ◽  
Red Flags ◽  
Fake News ◽  
Electoral Fraud ◽  
State Regulations ◽  
Electoral Laws ◽  
American Elections

The contemporary era raises a series of red flags about electoral integrity in America. Problems include plummeting public trust, exacerbated by President Trump’s claims of massive electoral fraud. Confidence in the impartiality and reliability of information from the news media has eroded. And Russian meddling has astutely exploited both these vulnerabilities, heightening fears that the 2016 contest was unfair. This book brings together a first-class group of expert academics and practitioners to analyze challenges facing contemporary elections in America. Contributors analyze evidence for a series of contemporary challenges facing American elections, including the weaknesses of electoral laws, overly restrictive electoral registers, gerrymandering district boundaries, fake news, the lack of transparency, and the hodgepodge of inconsistent state regulations. The conclusion sets these issues in comparative context and draws out the broader policy lessons for improving electoral integrity and strengthening democracy.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

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