This volume presents a collection of articles highlighting recent developments in commutative algebra and related non-commutative generalizations. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non-Noetherian ring theory, module theory and integer-valued polynomials along with connections to algebraic number theory, algebraic geometry, topology and homological algebra.
Most of the eighteen contributions are authored by attendees of the two conferences in commutative algebra that were held in the summer of 2016: "Recent Advances in Commutative Ring and Module Theory," Bressanone, Italy; "Conference on Rings and Polynomials" Graz, Austria. There is also a small collection of invited articles authored by experts in the area who could not attend either of the conferences. Following the model of the talks given at these conferences, the volume contains a number of comprehensive survey papers along with related research articles featuring recent results that have not yet been published elsewhere.
Marco Fontana is professor of algebra at the Università degli Studi "Roma Tre". His research interests lie in the areas of commutative ring theory and related topological aspects, with main focus on multiplicative ideal theory, Prüfer-like conditions and ideal factorizations, and Zariski-Riemann spaces of valuation domains.
Sophie Frisch is associate professor of mathematics at Technische Universität Graz, Austria. Her research interests are in commutative algebra and ring theory, including, but not limited to, polynomial mappings and integer-valued polynomials.
Sarah Glaz is professor of mathematics at the University of Connecticut. Her research interests lie in the areas of commutative ring theory and homological algebra, with main focus on non-Noetherian properties such as coherence, finite conductor, Gaussian, and Prüfer-like conditions of rings and their modules.
Francesca Tartarone is associate professor of algebra at the Università degli Studi "Roma Tre". Her research interests lie in the area of commutative ring theory with a particular focus on integer valued polynomial rings, star operations on ideals and multiplicative ideal theory properties of Prüfer-like domains.
Paolo Zanardo is professor of algebra at the Università di Padova. His research interests lie in the area of commutative rings and their modules, with main focus on modules over valuation domains, ideal class semigroups, algebraic entropies, fully inert Abelian Groups, and factorization of matrices over integral domains.
Preface.- 1 Reducing Fractions to Lowest Terms.- 2 Unique Factorization in Torsion-free Modules.- 3 n-Absorbing Ideals of Commutative Rings and Recent Progresses on Three Conjectures: A Survey.- 4 Embedding Dimension and Codimension of Tensor Products of Algebras Over a Field.- 5 Minimal Generating Sets for the D-Algebra Int(S,D).- 6 Algebraic Entropy in Locally Linearly Compact Vector Spaces.- 7 Commutative Rings Whose Finitely Generated Ideals are Quasi-Flat.- 8 Commutative Rings with a Prescribed Number of Isomorphism Classes of Minimal Ring Extensions.- 9 Applications of Multisymmetric Syzygies in Invariant Theory.- 10 Functorial Properties of Star Operations: New Developments.- 11 Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids.- 12 Corner?s Realization Theorems from the Viewpoint of Algebraic Entropy.- 13 Directed Unions of Local Quadratic Transforms of Regular Local Rings and Pullbacks.- 14 Divisorial Prime Ideals in Prüfer Domains.- 15 A gg-Cancellative Semistar Operation on an Integral Domain Need Not Be gh-Cancellative.- 16 Quasi-Prüfer Extensions of Rings.- 17 A Note on Analytically Irreducible Domains.- 18 Integer-valued Polynomials on Algebras: A Survey of Recent Results and Open Questions.
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