Rings, Monoids and Module Theory: AUS-ICMS 2020, Sharjah, United Arab Emirates, February 6–9 (Springer Proceedings in Mathematics & Statistics Book 382)

This book contains select papers on rings, monoids and module theory which are presented at the 3rd International Conference on Mathematics and Statistics (AUS-ICMS 2020) held at the American University of Sharjah, United Arab Emirates, from 6–9 February 2020. This conference was held in honour of the work of the distinguished algebraist Daniel D. Anderson. Many participants and colleagues from around the world felt it appropriate to acknowledge his broad and sweeping contributions to research in algebra by writing an edited volume in his honor. The topics covered are, inevitably, a cross-section of the vast expansion of modern algebra. The book is divided into two sections―surveys and recent research developments―with each section hopefully offering symbiotic utility to the reader. The book contains a balanced mix of survey papers, which will enable expert and non-expert alike to get a good overview of developments across a range of areas of algebra. The book is expected to be of interest to both beginning graduate students and experienced researchers. Preface 7 Students of Daniel D. Anderson 8 Contents 10 About the Editors 12 Dan Anderson and His Mathematics 14 References 18 Bounded and Finite Factorization Domains 19 1 Introduction 20 2 Preliminary 21 2.1 General Notation 21 2.2 Factorizations 21 3 Bounded and Finite Factorization Monoids 23 3.1 The Bounded Factorization Property 23 3.2 The Finite Factorization Property 24 4 Bounded and Finite Factorization Domains 26 4.1 Characterizations of BFDs and (Strong) FFDs 26 4.2 Some Relevant Classes of BFDs and FFDs 29 4.3 The D+M Construction 32 5 Subrings, Ring Extensions, and Localizations 36 5.1 Inert Extensions 36 5.2 Subrings 39 5.3 Ring Extensions and Overrings 41 5.4 Pullback Constructions 45 6 Polynomial-Like Rings 48 6.1 Bounded Factorization Subdomains of R[X] and R[[X]] 48 6.2 Finite Factorization Subdomains of R[X] and R[[X]] 53 6.3 Monoid Domains 58 6.4 Graded Integral Domains 63 7 Generalized Bounded and Finite Factorization Domains 64 References 68 Factorization and Irreducibility in Modules 70 1 Introduction 70 2 Notions of Irreducibility 72 3 Notions of Atomicity 88 References 97 On -potent Domains and -homogeneous Ideals 99 1 Introduction 99 1.1 Introduction to Star Operations 100 2 -homogeneous Ideals 101 3 What -homogeneous Ideals Can Do 105 3.1 -f-potent Domains 108 References 118 On the Set of Molecules of Numerical and Puiseux Monoids 120 1 Introduction 120 2 Preliminary 122 3 Molecules of Interval Numerical Monoids 123 4 Molecularity of the Class of Numerical Monoids 126 4.1 Atomic Classes of Puiseux Monoids 126 4.2 A Conjecture on Molecularity 127 5 Molecularity of Further Classes of Puiseux Monoids 129 5.1 Molecularity of mathcalC2 129 5.2 Molecularity of Non-Atomic Puiseux Monoids 131 6 Infinite Molecularity 131 References 134 Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from, II 135 1 Introduction 135 2 Results 136 3 Some Recent History and Some Open Questions 140 References 142 A Survey on EM Conditions 143 1 Introduction 143 2 Annihilating Content 144 3 EM-Rings 145 4 The Noetherian Case 146 5 Some Extensions of EM-Rings 147 6 Some Generalizations 148 References 150 Some Remarks on the D + M Construction 151 References 155 On a Problem About Lowest Terms Domains Posed by D. D. Anderson 156 1 Introduction 156 2 Basic Properties of D 157 3 Main Results 165 References 176 Regularity and Related Properties in Tensor Products of Algebras Over a Field 177 1 Introduction 177 2 Preliminaries 179 3 Tensor Products of Cohen–Macaulay k-Algebras 184 4 Regularity, Gorensteiness, and Complete Intersection in Tensor Products 189 5 Regularity of Tensor Products of Extension Fields and Applications 192 References 199 Tame-Wild Dichotomy for Commutative Noetherian Rings—A Survey 201 1 Introduction 201 2 Tameness 203 2.1 The Local Case 203 2.2 Global Tameness 204 3 A Dichotomy Theorem 205 4 Wildness 206 5 Non-wildness of Dedekind-Like Rings 209 5.1 Brauer Groups 209 6 What's Left? 211 6.1 Field Extensions and Division Algebras 212 6.2 Unsplit Dedekind-Like Rings 215 References 215 On the Characterization of τ(n)-Atoms 216 1 Introduction 216 2 Preliminaries 218 3 Structure of U'(n) 219 4 Tools for Determining τ(n)-irreducible Elements for U'(n) 223 5 Computational Results 228 6 Conclusion 235 References 235 Bounded Periodic Rings 236 1 Introduction and Fundamentals 236 2 The Main Results 238 References 243 On Gracefully and Harmoniously Labeling Zero-Divisor Graphs 244 1 Introduction 244 2 Background 246 2.1 Zero-Divisor Graphs 246 2.2 Graceful Labeling 247 2.3 Harmonious Labeling 248 3 Labeling Results on Zero-Divisor Graphs of Commutative Rings 249 4 Small Zero-Divisor Graphs 252 References 264 A Survey on Genus of Selected Graphs from Commutative Rings 266 1 Introduction 266 2 Genus Properties of Graphs 267 3 Zero-Divisor Graphs 268 4 Jacobson Graphs 274 5 Annihilator Graphs 277 6 Unit Graphs 280 7 Unitary Cayley Graphs and Cayley Sum Graphs 283 8 Generalized Unit and Unitary Cayley Graphs 286 References 288 A Computational Approach to Shephard Groups 291 1 Introduction 291 2 Preliminaries 292 2.1 Gröbner-Shirshov Bases 292 2.2 Complex Reflection Groups 294 3 The Shephard Groups of Tetrahedral Type 296 3.1 The Group G4 296 3.2 The Group G5 299 3.3 The Group G6 300 4 The Shephard Groups of Octahedral Type 301 4.1 The Group G8 301 4.2 The Group G9 302 4.3 The Group G10 304 4.4 The Group G14 305 5 The BMR Freeness Theorem for Hecke Algebras 307 6 The BMM Symmetrizing Trace Conjecture 307 References 308 BZS Near-Rings and Rings 310 1 Introduction 310 2 The BZS Property and Basic Results 312 3 More Results on BZS Near-Rings and Rings 313 4 Avenues for Future Work 316 References 317 This book contains select papers on rings, monoids and module theory which are presented at the 3rd International Conference on Mathematics and Statistics (AUS-ICMS 2020) held at the American University of Sharjah, United Arab Emirates, from 6-9 February 2020. This conference was held in honour of the work of the distinguished algebraist Daniel D. Anderson. Many participants and colleagues from around the world felt it appropriate to acknowledge his broad and sweeping contributions to research in algebra by writing an edited volume in his honor. The topics covered are, inevitably, a cross-section of the vast expanse that is modern algebra. The book is divided into two sections--surveys and recent research developments--with each section hopefully offering symbiotic utility to the reader. The book contains a balanced mix of survey papers, which will enable expert and non-expert alike to get a good overview of developments across a range of areas of algebra. The book is expected to be of interest to both beginning graduate students and experienced researchers. .