Strong atoms in monadically Krull monoids
type: Item
instance of: scholarly article
title: Strong atoms in monadically Krull monoids
zbMATH Open document ID: 1481.13036
author: Gerhard Angermüller
published in: Semigroup Forum
publication date: 2022-01-25T00:00:00Z
review text: This manuscript studies strong atoms on monadically Krull monoids, with some applications to the ring of integer-valued polynomials on integral domains. Monoids in this paper are considered to be commutative and cancellative. An atom in a monoid is an element that cannot be expressed as a product of two nonunit elements in the monoid. An atom \(a\) is strong if every divisor of a power of \(a\) is also a power of \(a\). A nonunit \(a\) in a monoid \(M\) is said to be prime if whenever \(a\) divides \(b c\), with \(b\) and \(c\) in \(M\), then either \(a\) divides \(b\) or \(c\), and it is said to be primary if whenever \(a\) divides \(b c\) and does not divide \(b\), then it divides some power of \(c\). For an element \(a\) in a monoid \(H\), the set \([\![a]\!]_H\) denotes the smallest divisor-closed submonoid of \(H\) containing \(a\). A property (P) is said to be monadic for \(H\) if (P) holds for \(H\) if and only if (P) holds for \([\![a]\!]_H\) for every \(a\in H\). A monoid \(H\) has property (S) if for every nonunit \(x\) of \(H\) there exists a finite set of strong atoms \(A\) such that (1) \(x^m = w \prod_{a\in A} a^{m_a}\) for some positive integers \(m\), \(m_a\), \(a\in A\), and some unit \(w\), (2) this expression is unique, and (3) if \(y\) is a strong atom dividing a power of \(x\), then \(y\in A\). The author shows that property (S) is monadic. The extraction degree of \(H\) for two elements \(x,y\in H\) is defined as \(\lambda_H(x,y)= \sup\{ m/n : m\in \mathbb{N}_0, n\in \mathbb{N}, x^m \mid_H y^n \}\). The monoid \(H\) is said to be an extraction monoid if for every nonunit \(x\) in \(H\) and every \(y\in H\), there exists a nonnegative integer \(m\) and a positive integer \(n\) such that \(\lambda_H(x,y)=m/n\). The property of being an extraction monoid is also monadic. A monoid \(H\) is locally finitely generated if for every \(a\in A\), \(([\![a]\!])_H)_{red}\) is finitely generated (where the subindex ''red'' stands for reduced, ''units remove…
reviewed by: Pedro A. García Sánchez
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M13
zbMATH DE Number: 7463821
zbMATH Keywords: absolutely irreducible
zbMATH Keywords: integer-valued polynomial
zbMATH Keywords: extraction monoid
zbMATH Keywords: Krull monoid
zbMATH Keywords: monadically Krull monoid
zbMATH Keywords: strong atom
MaRDI profile type: Publication
full work available at URL: s00233-021-10231-9
OpenAlex ID: W3210204737
cites work: Roots in extensions of domains or monoids
cites work: Strong atoms in Krull monoids
cites work: Q5625315
cites work: Inside factorial monoids and integral domains
cites work: Rings, Modules, and Closure Operations
cites work: Relative Polynomial Closure and Monadically Krull Monoids of Integer-Valued Polynomials
cites work: A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator
cites work: Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields
cites work: Q3436766
cites work: Q5661989
cites work: Non-absolutely irreducible elements in the ring of integer-valued polynomials
cites work: On Monoids and Domains Whose Monadic Submonoids Are Krull
cites work: ON THE DIVISOR-CLASS GROUP OF MONADIC SUBMONOIDS OF RINGS OF INTEGER-VALUED POLYNOMIALS
cites work: Absolute irreducibility of the binomial polynomials
DOI: 10.1007/S00233-021-10231-9
description: scientific article; zbMATH DE number 7463821