The Ideal Theory and Arithmetic of Rings, Monoids, and Semigroups

Scott T Chapman

• Publisher: American Mathematical Society
• Year: 2026
• Topic: Mathematics
• eBook ISBN:
  9781470485443
• Subtopic: Mathematics General
• Index: Mathematics

This volume is based on talks given at the Special Session at the second International Joint Meeting of the Unione Matematica Italiana and the American Mathematical Society, held at the Universitá degli Studi di Palermo on July 23–26, 2024. During the last twenty years, the theory involving the structure of the arithmetic and ideal theory of various algebraic structures has been a popular topic and taken several important steps forward. Many applications of this theory, with particular attention to the multiplicative monoids of integral domains and their combinatorial or numerical applications to ring theory, have appeared throughout the mathematical literature. The aim of this volume is to review recent developments in this area by bringing together researchers from different areas of algebra under the umbrella of commutative monoids, semigroups, and rings. Topics include multiplicative ideal theory and general ideal systems, arithmetic in Krull and Prüfer monoids, commutative monoid rings, integer-valued polynomials, numerical and congruence monoids, direct sum decompositions of modules, and various aspects of non-unique factorization.

Table of contents

• Cover
• Title page
• Contents
• Preface
• On long-term problems in multiplicative ideal theory and factorization theory
• Extremal elasticity of quadratic orders
• On the smallest partition associated to a numerical semigroup
• When the divisibility poset of the ideal class monoid of a numerical semigroup is a lattice
• On numerical semigroup elements and the ℓ₀ and ℓ_{∞} norms of their factorizations
• On three families of dense Puiseux monoids
• The set of minimal distances of the monoid of plus-minus weighted zero-sum sequences and applications to the characterization 瀀爀漀戀氀攀洀
• On counterexamples to Mordell’s Pellian Equation Conjecture and the AAC Conjecture: A non-computer-based approach
• A generalized Davenport constant of the second kind
• Additive subgroups of a module that are saturated with respect to a subset of the ring
• The 𝑛-total graph of a commutative ring
• Radii of convergence of algebraic power series
• A length function of ℤ[𝕏₁,…,𝕏ₘ]-modules and Mahler measure
• Weakly Arf property for quadratic quotients of the Rees algebra