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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.