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 Students of Daniel D. Anderson Contents About the Editors Dan Anderson and His Mathematics References Bounded and Finite Factorization Domains 1 Introduction 2 Preliminary 2.1 General Notation 2.2 Factorizations 3 Bounded and Finite Factorization Monoids 3.1 The Bounded Factorization Property 3.2 The Finite Factorization Property 4 Bounded and Finite Factorization Domains 4.1 Characterizations of BFDs and (Strong) FFDs 4.2 Some Relevant Classes of BFDs and FFDs 4.3 The D+M Construction 5 Subrings, Ring Extensions, and Localizations 5.1 Inert Extensions 5.2 Subrings 5.3 Ring Extensions and Overrings 5.4 Pullback Constructions 6 Polynomial-Like Rings 6.1 Bounded Factorization Subdomains of R[X] and R[[X]] 6.2 Finite Factorization Subdomains of R[X] and R[[X]] 6.3 Monoid Domains 6.4 Graded Integral Domains 7 Generalized Bounded and Finite Factorization Domains References Factorization and Irreducibility in Modules 1 Introduction 2 Notions of Irreducibility 3 Notions of Atomicity References On -potent Domains and -homogeneous Ideals 1 Introduction 1.1 Introduction to Star Operations 2 -homogeneous Ideals 3 What -homogeneous Ideals Can Do 3.1 -f-potent Domains References On the Set of Molecules of Numerical and Puiseux Monoids 1 Introduction 2 Preliminary 3 Molecules of Interval Numerical Monoids 4 Molecularity of the Class of Numerical Monoids 4.1 Atomic Classes of Puiseux Monoids 4.2 A Conjecture on Molecularity 5 Molecularity of Further Classes of Puiseux Monoids 5.1 Molecularity of mathcalC2 5.2 Molecularity of Non-Atomic Puiseux Monoids 6 Infinite Molecularity References Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from, II 1 Introduction 2 Results 3 Some Recent History and Some Open Questions References A Survey on EM Conditions 1 Introduction 2 Annihilating Content 3 EM-Rings 4 The Noetherian Case 5 Some Extensions of EM-Rings 6 Some Generalizations References Some Remarks on the D + M Construction References On a Problem About Lowest Terms Domains Posed by D. D. Anderson 1 Introduction 2 Basic Properties of D 3 Main Results References Regularity and Related Properties in Tensor Products of Algebras Over a Field 1 Introduction 2 Preliminaries 3 Tensor Products of Cohen–Macaulay k-Algebras 4 Regularity, Gorensteiness, and Complete Intersection in Tensor Products 5 Regularity of Tensor Products of Extension Fields and Applications References Tame-Wild Dichotomy for Commutative Noetherian Rings—A Survey 1 Introduction 2 Tameness 2.1 The Local Case 2.2 Global Tameness 3 A Dichotomy Theorem 4 Wildness 5 Non-wildness of Dedekind-Like Rings 5.1 Brauer Groups 6 What's Left? 6.1 Field Extensions and Division Algebras 6.2 Unsplit Dedekind-Like Rings References On the Characterization of τ(n)-Atoms 1 Introduction 2 Preliminaries 3 Structure of U'(n) 4 Tools for Determining τ(n)-irreducible Elements for U'(n) 5 Computational Results 6 Conclusion References Bounded Periodic Rings 1 Introduction and Fundamentals 2 The Main Results References On Gracefully and Harmoniously Labeling Zero-Divisor Graphs 1 Introduction 2 Background 2.1 Zero-Divisor Graphs 2.2 Graceful Labeling 2.3 Harmonious Labeling 3 Labeling Results on Zero-Divisor Graphs of Commutative Rings 4 Small Zero-Divisor Graphs References A Survey on Genus of Selected Graphs from Commutative Rings 1 Introduction 2 Genus Properties of Graphs 3 Zero-Divisor Graphs 4 Jacobson Graphs 5 Annihilator Graphs 6 Unit Graphs 7 Unitary Cayley Graphs and Cayley Sum Graphs 8 Generalized Unit and Unitary Cayley Graphs References A Computational Approach to Shephard Groups 1 Introduction 2 Preliminaries 2.1 Gröbner-Shirshov Bases 2.2 Complex Reflection Groups 3 The Shephard Groups of Tetrahedral Type 3.1 The Group G4 3.2 The Group G5 3.3 The Group G6 4 The Shephard Groups of Octahedral Type 4.1 The Group G8 4.2 The Group G9 4.3 The Group G10 4.4 The Group G14 5 The BMR Freeness Theorem for Hecke Algebras 6 The BMM Symmetrizing Trace Conjecture References BZS Near-Rings and Rings 1 Introduction 2 The BZS Property and Basic Results 3 More Results on BZS Near-Rings and Rings 4 Avenues for Future Work References 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. .