Advances in Rings, Modules and Factorizations: Graz, Austria, February 19-23, 2018 (Springer Proceedings in Mathematics & Statistics Book 321)

Occasioned by the international conference "Rings and Factorizations" held in February 2018 at University of Graz, Austria, this volume represents a wide range of research trends in the theory of commutative and non-commutative rings and their modules, including multiplicative ideal theory, Dedekind and Krull rings and their generalizations, rings of integer valued-polynomials, topological aspects of ring theory, factorization theory in rings and semigroups and direct-sum decompositions of modules. The volume will be of interest to researchers seeking to extend or utilize work in these areas as well as graduate students wishing to find entryways into active areas of current research in algebra. A novel aspect of the volume is an emphasis on how diverse types of algebraic structures and contexts (rings, modules, semigroups, categories) may be treated with overlapping and reinforcing approaches. Preface Contents Commutative Rings Whose Principal Ideals Have Unique Generators 1 Introduction 2 The Quasivariety of Rings Whose Principal Ideals have Unique Generators 3 Some Related Quasivarieties References On Monoids of Ideals of Orders in Quadratic Number Fields 1 Introduction 2 Preliminaries on the Arithmetic of Monoids 2.1 Monoids 2.2 Sets of Lengths 2.3 Distances and Chains of Factorizations 2.4 Semigroups of Ideals 3 Orders in Quadratic Number Fields 4 Sets of Distances and Sets of Catenary Degrees 5 Unions of Sets of Lengths References UMT-domains: A Survey 1 Introduction 2 Definitions Related to the t-operation 2.1 Star Operations 2.2 *-Invertibility 2.3 t-Class Group and Picard Group 2.4 PvMDs, Nagata Rings, and Kronecker Function Rings 3 UMT-domains 3.1 UMT-domains 3.2 Kaplansky Type Theorems 4 Graded Integral Domains Which Are UMT-domains 4.1 Definitions of Graded Integral Domains 4.2 General Case 4.3 When RH Is a UFD 5 Constructing New UMT-domains from Old Ones 5.1 Semigroup Rings 5.2 Pullbacks References The Apéry Set of a Good Semigroup 1 Introduction 2 Preliminaries 3 Apéry Set of Good Semigroups in mathbbN2 4 Symmetric Good Semigroups 5 Duality of the Apéry Set of Symmetric Good Semigroups References On Syzygies for Rings of Invariants of Abelian Groups 1 Introduction 2 Ring Theoretic Characterization of the Catenary Degree 3 Graded Monoids 4 Repetition of Elements 5 Gröbner Bases 6 Relation to Invariant Theory 7 Relation to Toric Varieties References A Bazzoni-Type Theorem for Multiplicative Lattices 1 Introduction 2 Main Result 3 Almost Prüfer Extensions 4 Ideal Systems on Monoids and Integral Domains References What is the Spectral Category? 1 Introduction 2 Stabilization of Classes of Morphisms 3 Essential and Pullback Stable Essential Monomorphisms 4 Spans and Fractions 5 The Spectral Category 6 Subobject-Essential Monomorphisms 7 Uniform Objects References A Survey on the Local Invertibility of Ideals in Commutative Rings 1 Introduction 2 The Prüfer Case 3 Generalization to Non-Prüfer Domains 4 Generalization to Rings with Zero-Divisors References Idempotence and Divisoriality in Prüfer-Like Domains 1 Introduction and Preliminaries 2 Idempotence 3 Divisoriality References Simultaneous Interpolation and P-adic Approximation by Integer-Valued Polynomials 1 Introduction 2 Notation and Definitions 3 P-adic Lipschitz Constants of Interpolating Integer-Valued Polynomials 4 Interpolation by Congruence-Preserving Integer-Valued Polynomials References Length Functions over Prüfer Domains 1 Introduction 2 Generalities on Length Functions 3 Length Functions over Valuation Domains 4 Properties of Prüfer Domains and Their Meet-Spectrum 5 Length Functions over Semilocal Prüfer Domains 6 Length Functions over Prüfer Domains of Finite Character References On Some Arithmetical Properties of Noetherian Domains 1 Introduction 2 Quasi Finitely Generated Monoids 3 Root Extensions of Noetherian Rings References Spectral Spaces Versus Distributive Lattices: A Dictionary 1 Distributive Lattices and Spectral Spaces: Some General Facts 1.1 The Seminal Paper by Stone 1.2 Distributive Lattices and Entailment Relations 2 Spectral Spaces in Algebra 2.1 Dynamical Algebraic Structures, Distributive Lattices and Spectra 2.2 A Very Simple Case 2.3 The Real Spectrum of a Commutative Ring 2.4 Linear Spectrum of a Lattice-Group 2.5 Valuative Spectrum of a Commutative Ring 2.6 Heitmann Lattice and J-Spectrum of a Commutative Ring 3 A Short Dictionary 3.1 Properties of Morphisms 3.2 Dimension Properties 3.3 Properties of Spaces 4 Some Examples 4.1 Relative Dimension, Lying Over, Going Up, Going Down 4.2 Kronecker, Forster–Swan, Serre and Bass Theorems 4.3 Other Results Concerning Krull Dimension References Valuative Marot Rings 1 Introduction 2 Preliminary Results 3 Valuative Marot Rings References Classifying Modules in Add of a Class of Modules with Semilocal Endomorphism Rings 1 Overview of the Finite Case 2 I-small Modules 3 The Infinite Case References The Multiplicative Ideal Theory of Leavitt Path Algebras of Directed Graphs—A Survey 1 Introduction 2 Preliminaries 3 Graded Ideals of a Leavitt Path Algebra 4 The Lattice of Ideals of a Leavitt Path Algebra 5 Prime, Radical, Primary, and Irreducible Ideals of a Leavitt Path Algebra 6 Factorization of Ideals in L References When Two Principal Star Operations Are the Same 1 Introduction 2 Background 3 Principal Star Operations 4 Local Rings 5 Localizations 6 v-Trivial Ideals 7 Noetherian Domains References Tilting Modules and Tilting Torsion Pairs 1 Introduction 2 Classical n-tilting Modules 3 First Attempts to Recover the Decomposition 4 Introducing Derived Categories and t-structures 5 n-Tilting Objects and Associated t-structures 6 The t-tree References