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 Universita 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 Prufer monoids, commutative monoid rings, integer-valued polynomials, numerical and congruence monoids, direct sum decompositions of modules, and various aspects of non-unique factorization.
Alfred Geroldinger, Hwankoo Kim, and K. Alan Loper,
On long-term problems in multiplicative ideal theory and factorization
theory;
Kai Steve Fan and Paul Pollack,
Extremal elasticity of quadratic orders;
Nathan Kaplan, Kaylee Kim, Cole McGeorge, Fabian Ramirez, and Deepesh
Singhal,
On the smallest partition associated to a numerical semigroup;
S. Bonzio and P. A. Garcia-Sanchez,
When the divisibility poset of the ideal class monoid of a numerical
semigroup is a lattice;
Sogol Cyrusian, Alex Domat, Christopher O'Neill, Vadim Ponomarenko, Eric Ren,
and Mayla Ward,
On numerical semigroup elements and the $\ell_0$ and $\ell_\infty$ norms of
their factorizations;
Scott T. Chapman, Felix Gotti, Marly Gotti, and Harold Polo,
On three families of dense Puiseux monoids;
Kamil Merito, Oscar Ordaz, and Wolfgang A. Schmid,
The set of minimal distances of the monoid of plus-minus weighted zero-sum
sequences and applications to the characterization problem;
Andreas Reinhart,
On counterexamples to Mordell's Pellian Equation Conjecture and the AAC
Conjecture: A non-computer-based approach;
Jared Kettinger, A generalized Davenport constant of the second kind;
Jesse Elliott and Neil Epstein,
Additive subgroups of a module that are saturated with respect to a subset of
the ring;
Djamila AitElhadi and Ayman Badawi,
The $n$-total graph of a commutative ring;
Joseph Swanson, Radii of convergence of algebraic power series;
Davide Castelnovo, Dikran Dikranjan, Anna Giordano Bruno, Dario Spirito, and
Simone Virili,
A length function of $\mathbb{Z}[ X_1\ldots,X_m]$-modules and Mahler measures;
Damiano Saccone, Weakly Arf property for quadratic quotients of the Rees
algebra
Scott T. Chapman, Sam Houston State University, Huntsville, TX
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