Algebraic, Number Theoretic, and Topological Aspects of Ring Theory

Preface
Contents
Paul-Jean Cahen (1946–2019)
References
Paul-Jean Cahen's Mathematical Publications
Bhargava's Exponential Functions and Bernoulli Numbers Associated to the Set of Prime Numbers
1 Introduction
2 Bhargava's Exponential Functions
3 The Factorial Sequence of the Set of Prime Numbers
4 The Exponential Function Associated to P
5 Bernoulli Numbers Associated to P
6 Generalized Bernoulli Polynomials
7 A Subset E Such That k!E=(k+1)!E for Infinitely Many k
References
Polynomial Root Extensions
1 Introduction
2 The S-Closure
3 The S-Closure
Appendix
Dan Anderson (1948–2022) by David F. Anderson
References
Absorbing Ideals in Commutative Rings: A Survey
1 Introduction
2 Conjectures on n-Absorbing Ideals of Commutative Rings
3 2-AB-Rings and Factorization Rings
4 Commutative Rings with 2-Absorbing Factorization
5 Commutative Rings with Absorbing Factorization
References
Complement-Finite Ideals
1 Introduction and Motivation
2 Background and Terminology
3 Additional Structure and Algebraic Properties
3.1 Restrictions of Primes
3.2 Algebraic Properties
4 Atoms and Arithmetic
4.1 Complement-Finite Ideals Whose Minimal Elements Lie on Extremal Rays
5 Sequences Over a Finite Abelian Group That Are Not Zero-sum Free
Appendix
Nick Baeth (1978–2021) by Scott Chapman and James B. Coykendall
References
When Is a Group Algebra Antimatter?
1 Introduction
2 Antimatter Group Algebras with Rational Exponents
3 Antimatter Group Algebras
References
Yosida, Martínez, and A+B Rings
1 Introduction
2 A+B Rings
3 Lattice-Ordered Groups
4 AQB
References
Functional Identities and Maps Preserving Two-Sided Zero Products
1 Introduction
2 Functional Identities
3 Maps Preserving Two-Sided Zero Products
References
Bounded Factorization and the Ascending Chain Condition on Principal Ideals in Generalized Power Series Rings
1 Introduction
2 Background
3 Présimplifiability and Related Notions in Generalized Power Series Rings
4 Bounded Factorization and the ACCP in Generalized Power Series Rings
References
Probabilities and Fixed Divisors of Integer Polynomials
1 Turk's Formula
2 Kempner-Bhargava's Formula
3 ``Prob'' (f(S)I fOK[X], degf ≤d)
4 ``Prob''(d(S,f)= I f OK[X], degf≤d)
5 ``Prob''(d(S,f)= OK fOK [X], degf≤d)
References
Modules over Trusses vs. Modules over Rings: Internal Direct Sums
1 Introduction
2 Preliminaries
2.1 Heaps
2.2 Trusses
2.3 Modules over Trusses
3 Main Results
3.1 Internal Direct Sums of Modules over Trusses
3.2 Modules over Trusses vs. Modules over Rings
References
A Survey on Essential-Like Properties of Prüfer v-Multiplication Domains
Introduction
1 Some General Properties and the t-Finite Character
2 Integer-Valued Polynomials
References
On the Subatomicity of Polynomial Semidomains
1 Introduction
2 Background
2.1 Monoids
2.2 Semirings
3 Furstenbergness
4 Almost Atomicity
5 Quasi-atomicity
References
Invertibility, Semistar Operations, and the Ring of Finite Fractions
References
The Quadratic Tree of a Two-Dimensional Regular Local Ring
1 Introduction
2 Local Quadratic Transforms
3 The Quadratic Tree
4 The Topology of the Quadratic Tree
5 Intersections of Rings in the Quadratic Tree
6 Projective Models and the Quadratic Tree
References
Reductions and Core of Ideals in Integral Domains: Some Recent Developments
1 Introduction
2 Core of Ideals in Integral Domains
2.1 Core of Ideals
2.2 Minimal Reductions
3 Core of Ideals in One-Dimensional Noetherian Domains
3.1 Two Results
3.2 Illustrative Examples
4 Reductions of Ideals in Pullbacks
5 Minimal Reductions and Core of Ideals in Pullbacks
5.1 Minimal Reductions
5.2 Core
References
References
Valuative Lattices and Spectra
1 Introduction
2 Distributive Lattices and Spectral Spaces
2.1 The Seminal Paper by Stone
Ideals and Filters in a Distributive Lattice
The Spectrum of a Distributive Lattice
Stone's Antiequivalence
Finite Spectral Spaces
2.2 Distributive Lattices and Entailment Relations
2.3 Gluing Distributive Lattices and Spectral Subspaces
Quotients, Covers, Gluing Procedures
The Dual Viewpoint
2.4 Short Dictionary of Stone's Antiequivalence
Properties of Morphisms
Dimension Properties
Properties of Spaces
3 Dynamical theories and dynamical algebraic structures
3.1 Finitary Dynamical Theories
Collapsus
Classification of Dynamical Rules
A Basic Example
3.2 Dynamic Algebraic Structures
Constructive Models Versus Classical Models
3.3 Conservative Extensions
Essentially Identical Extensions
Essentially Equivalent Extensions
Other Conservative Extensions
4 Distributive lattices associated to a dynamical algebraic structure
4.1 Zariski Spectrum and Zariski Lattice of a Commutative Ring
4.2 Real lattice of a commutative ring
4.3 Other Examples
4.4 The Absolute Zariski Lattice of a Dynamical Algebraic Structure A
4.5 Spectrum and Models in Classical Mathematics
5 Valuative Lattice and Spectrum of a Commutative Ring
5.1 Valuative Divisibility Relation
Valuation Domains, Valuative Divisibility Relation
Points of the Valuative Spectrum in Classical Mathematics
5.2 Weakly Disjunctive Theories for a Valuative Divisibility Relation
The Theory val0
The Theory val and Some Dynamical Rules Provable in It
Dynamical Algebraic Structures of Type val
5.3 Valuative Lattice and Spectrum of a Commutative Ring
Several Possible Spectral Topologies
The Lattice val(A) and Its Spectrum Spev A
The Lattice val(A) and the Spectrum Spv A
5.4 Valuative Lattice and Spectrum of an Algebra
The Lattice val(A,k) and the Spectrum Spev(A,k)
The Center Map (1)
5.5 The Theory Val and the Lattice Val(K,k)
The Theory Val+
Isomorphism of Lattices val(K,k) and Val(K,k)
6 Valuative Dimensions
6.1 pp-Rings
6.2 The Ring Amin
6.3 Three Constructive Versions of Valuative Dimensions
7 Comparisons with a Theory of Valued Discrete Fields
7.1 Introduction
7.2 The Theory Vdf of Valued Discrete Fields
7.3 Formal Valuativstellensatz for Vdf and Consequences
7.4 Formal Valuativstellensätze for val and Vdf+
A Formal Valuativstellensatz for val(A)
Admissibility of the Rule DIV for the Weakly Disjunctive Theory val
A Formal Valuativstellensatz for val(A,k)
The Center Map (2)
Another Formal Valuativstellensatz for val(A,k)
References
Building Three-Variable Homogeneous Integer-Valued Polynomials Using Generalized Projective Planes
1 Introduction
2 Projective Planes over Finite Fields
3 Finite Projective Hjelmslev Planes
References
Around Prüfer Extensions of Rings
1 Introduction and Notation
1.1 An Overview of the Paper
1.2 Basics Concepts
2 Some Definitions, Notation, and Useful Results
3 S-Regular Ideals and Rings of Sections
4 Integral Closures as Intersections
5 Avoidance Lemmata
6 Pullback Results
7 The Case of a Local Base Ring
8 QR Extensions
9 Prüfer FCP Extensions over a Local Ring
10 The Set of All Primitive Elements
References
A Survey on Algebraic and Homological Properties of Amalgamated Algebras of Commutative Rings
1 Introduction
2 Algebraic Properties
2.1 Prime Ideals and Localization
2.2 Zero-Divisors
2.3 Integral Domain and Reduced Property
2.4 Noetherian and Coherent Property
2.5 Embedding Dimension
3 Cohen-Macaulay and Gorenstein Property
3.1 Cohen-Macaulay Property
3.2 Serre's Conditions
3.3 Gorenstein and Quasi-Gorenstein Properties
3.4 Cohen-Macaulay Property in the Sense of Ideals
3.5 Cohen-Macaulay Property in the Sense of Hamilton and Marley
4 Clean Property and Prüfer-Like Conditions
4.1 Clean Property
4.2 Prüfer-Like Conditions
5 Other Properties
References
The Ring of Integer-Valued Polynomials on 33 Matrices and Its Integral Closure
1 Introduction
2 The Maximal Order 3
2.1 Conjugacy Classes of 3 Modulo π
2.2 Decomposition of T
3 The ν-Sequence of 3
3.1 Characteristic Polynomials of Subsets of 3
4 Toward Computing ν-Sequences
4.1 Characteristic Polynomials for Elements in S
4.2 Characteristic Polynomials for Elements in T2
4.3 Characteristic Polynomials for Elements in T4
5 A Regular Basis for 3
References
Simultaneous p-Orderings and Equidistribution
1 Introduction
1.1 Integer-Valued Polynomials and Test Sets
1.2 Equidistribution and Simultaneous p-Orderings
1.3 Notation
1.4 Structure of the Paper
2 n-Optimal Sets for Quadratic Imaginary Number Fields
3 Estimate on the Energy of n-Optimal Sets
4 n-Optimal Sets for an Arbitrary Number Field
4.1 Enclosure of n-Optimal Sets in Cylinders
4.2 Limit Measures
4.3 Discrepancy
4.4 Proof of Theorem 2
5 Open Problems
5.1 Function Fields
5.2 Schinzel's Problem
References
A Survey on Flatness in Integer-Valued Polynomial Rings
1 Introduction
2 Flatness of Int(D), or More Generally Int(E,D), as a D-Module
3 Flatness of Int(D) as an Overring of D[X]
4 Some Illustrating Examples
4.1 Examples of Integral Domains D Such That Int(D) Is Either Locally Free or Faithfully Flat Over D
4.2 Examples of Integral Domains D Such That Int(D) Is Not Flat Over D[X]
4.3 Examples of Integral Domains D Such That Int(D) Is Flat Over D But Not Over D[X]
References
Equivalent Characterizations of Non-Archimedean Uniform Spaces
1 Introduction
2 Equivalent Approaches to Non-Archimedean Uniform Spaces
2.1 The General Case
2.2 The Non-Archimedean Case
3 An Equivalent Separation Axiom
3.1 The General Case
3.2 The Non-Archimedean Case
4 Pseudo-Metrizability
4.1 The General Case
4.2 The Non-Archimedean Case
References