Multiplicative Ideal Theory and Factorization Theory (eBook)

Contents 6
Introduction 8
The Conference in Graz 8
A Tribute to Franz Halter-Koch 10
References 12
Multiplicative Ideal Theory in Noncommutative Rings 16
1 Introduction 16
2 Maximal Orders 17
3 Krull Orders 21
4 Unique Factorization Rings 25
5 G-Dedekind Prime Rings 29
6 Hereditary Noetherian Prime Rings (HNP Rings) and a Generalization of HNP Rings 31
References 34
About Number Fields with Pólya Group of Order leq2 37
1 Introduction 37
2 The Hypothesis Card(mathcalPo(K)) leq2 39
3 Toward Reciprocal Assertions 41
3.1 Counterexamples 41
3.2 Nontrivial Relations in mathcalPO(K) 42
3.3 Factorizations in Monoids 44
3.4 Tame Ramification 45
4 Galois Number Fields of Prime Degree 46
5 Quadratic Number Fields 49
6 A Few Words About the Function Fields Case 51
References 56
The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics 57
1 Introduction 57
2 Multiplicative Ideal Theory: Krull Monoids, C-Monoids, and Class Groups 60
2.1 Monoids and Domains: Ideal Theoretic and Divisor Theoretic Concepts 61
2.2 Class Groups and Class Semigroups 62
2.3 Krull Monoids and Krull Domains 63
2.4 C-Monoids and C-Domains 65
2.5 Davenport Constants of BF-Monoids 67
3 Arithmetic Combinatorics: Zero-Sum Results with a Focus on Davenport Constants 71
3.1 The Monoid of Product-One Sequences 71
3.2 Transfer Homomorphisms 77
3.3 The kth Davenport Constants: The General Case 79
3.4 The kth Davenport Constants: The Abelian Case 82
4 Multiplicative Ideal Theory of Invariant Rings 86
4.1 Basics of Invariant Theory 86
4.2 The Divisor Theory of Invariant Rings 89
4.3 The Abelian Case 93
4.4 A Monoid Associated with G-Modules 98
5 Constants from Invariant Theory and Their Counterparts in Arithmetic Combinatorics 101
5.1 The Noether Number Versus the Davenport Constant 101
5.2 The Constants ?(G, V) and ?(G,V) 105
References 106
Ring and Semigroup Constructions 110
1 Introduction 110
2 Survey on Ring Constructions 112
3 Semigroup Constructions 118
3.1 Algebroid Branches 120
3.2 Symmetry 124
References 127
New Distinguished Classes of Spectral Spaces: A Survey 129
1 Introduction and Preliminaries 130
1.1 Semistar Operations 130
1.2 Riemann--Zariski Spaces 131
1.3 The Inverse Topology on a Spectral Space 131
2 Ultrafilter Topology and Spectral Spaces 132
3 Spaces of Semistar Operations 138
4 The Space of Inverse-Closed Subsets of a Spectral Space 145
5 A Topological Version of Hilbert's Nullstellensatz 148
6 The Space of eab Semistar Operations of Finite Type 150
References 153
Relative Polynomial Closure and Monadically Krull Monoids of Integer-Valued Polynomials 156
1 Introduction 156
2 Relative Polynomial Closure 159
3 Finite Polynomially Dense Subsets 160
4 Divisor Theories for Monoids of Integer-Valued Polynomials on Discrete Valuation Rings 162
5 Divisor Homomorphisms on Monadic Monoids of Integer-Valued Polynomials 165
References 168
An Overview of the Computational Aspects of Nonunique Factorization Invariants 169
1 Introduction 169
2 Presentations 173
3 Apéry Sets 175
4 Graver Bases 175
5 Block Monoids 177
6 Denumerant and Maximal Denumerant 178
7 Length-Based Invariants 178
7.1 Elasticity 179
7.2 Delta Sets 180
8 Distance-Based Invariants 181
8.1 Catenary Degree 182
8.2 Monotone, Equal, and Homogeneous Catenary Degrees 184
8.3 Tame Degree 186
9 ?-Primality 188
References 189
Arithmetic of Mori Domains and Monoids: The Global Case 192
1 Introduction 192
2 Generalities on Semigroups and Monoids 193
3 Class Semigroups 196
4 Quasi-Finite Semigroups 201
5 Weakly C-Monoids 207
6 Arithmetic of Weakly C-Monoid 214
7 Integral Domains 219
References 226
Prüfer Domains of Integer-Valued Polynomials 228
1 Introduction 228
2 Int(D) 229
2.1 Noetherian Domains 229
2.2 Non-Noetherian Domains 230
3 Int(E, D) 237
4 Generalizations 238
References 240
Lobal Properties of Integral Domains 241
1 Introduction 241
2 Prime and Primary Ideals 244
3 Invertible Ideals, Coherent Domains, H-Local Domains and PVDs 247
4 UFDs, Krull Domains, Atomic Domains, and HFDs 254
References 262
Noetherian Semigroup Algebras and Beyond 263
1 Introduction 263
2 Group Algebras---Introductory Results 264
3 A General Structural Approach 267
4 Important Motivating Examples---Algebras with Homogeneous Quadratic Relations 270
5 Prime Ideals and Arithmetical Properties of K[S] 272
6 Why Should We Look at the Non-Noetherian Case? Motivation and First Results 279
References 282
Topological Aspects of Irredundant Intersections of Ideals and Valuation Rings 285
1 Introduction 285
2 Spectral Spaces 287
3 Spectral Representations 292
4 Irredundance in Intersections of Valuation Rings 298
5 Generalizations of Krull Domains 304
6 Prüfer v-.4-Multiplication Domains 306
7 Irredundance in Intersections of Irreducible Ideals 312
References 313
Idempotent Pairs and PRINC Domains 316
1 Introduction 317
2 Ideals Generated by an Idempotent Pair 318
3 Orders in Number Fields and Idempotent Pairs 321
4 mathbbZ[sqrt- d] Not Satisfying (Princ) 324
5 mathbbZ[sqrt-3] and mathbbZ[sqrt-7] are PRINC Domains 325
References 328
Some Recent Results and Open Problems on Sets of Lengths of Krull Monoids with Finite Class Group 330
1 Introduction 330
2 Preliminaries 331
2.1 Monoids, Factorizations, Sets of Lengths 331
2.2 Abelian Groups and Zero-Sum Sequences 332
2.3 Krull Monoids and Transfer Homomorphisms 334
2.4 Examples of Krull Monoids and Related Structures 335
3 Some General Results 336
4 Small Sets 337
5 Distances 344
6 Large Sets 348
6.1 The Relevance of AAMPs 350
6.2 Some Special Cases 351
6.3 The Set of Differences 352
6.4 The Bound in the STSL 356
References 358
Factorizations of Elements in Noncommutative Rings: A Survey 360
1 Introduction 360
2 Preliminaries 362
2.1 Small Categories as Generalizations of Semigroups 363
2.2 Classical Maximal Orders 364
3 Factorizations and Atomicity 367
3.1 Rigid Factorizations 367
3.2 Factor Posets 369
3.3 Atomicity, BF-Categories, and FF-Categories 370
4 Unique Factorization 372
4.1 Similarity Factorial Domains and Related Notions 372
4.2 A Different Notion of UFRs and UFDs 385
5 Nonunique Factorizations 389
5.1 Arithmetical Invariants 389
5.2 FF-Domains 392
5.3 Transfer Homomorphisms 394
5.4 Transfer Results 397
6 Other Results 405
References 405
Index 410