2019/10/31 by Alfred Geroldinger, Felix Gotti, Salvatore Tringali · 1 citation
Mathematics · #Algebra over a field #Algebraic number #Characterization (materials science) #Commutative Algebra and Its Applications #Factorization #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Monoid #Multiplicative function #Noetherian #Pure mathematics #Rational number #Rings, Modules, and Algebras #Unique factorization domain #math.AC #msc:13A05 #msc:20M13 #msc:20M14
paper · pdf · doi:10.1016/j.jalgebra.2020.09.019
published as J. Algebra 567 (2021), No. 1, pp. 310-345 · 25 pages. It will appear in Journal of Algebra
arxiv created 2020/09/25 · openalex publication_date 2020/09/29 · arxiv updated 2022/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. It is well-known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is Noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux monoids are primary monoids, and we provide conditions ensuring that they are strongly primary. Then we study local and global tameness of strongly primary Puiseux monoids; most notably, we establish an algebraic characterization of when a Puiseux monoid is globally tame. Moreover, we obtain a result on the structure of sets of lengths of all locally tame strongly primary monoids.