On the arithmetic of Krull monoids with finite Davenport constant.
type: Item
instance of: scholarly article
title: On the arithmetic of Krull monoids with finite Davenport constant.
zbMATH Open document ID: 1196.20066
author: Alfred Geroldinger
author: David J. Grynkiewicz
published in: Journal of Algebra
publication date: 2009-04-29T00:00:00Z
review text: The authors study factorization properties of Krull monoids \(H\) (i.e. commutative cancellative semigroups with divisor theory) in which the set \(G_P\) of classes containing prime divisors has a finite Davenport constant. It has been known that if \(G_P\) is finite, then for \(h\in H\) the set of lengths of factorizations of \(h\) forms an almost arithmetical multiprogression having uniformly bounded length. The main result of the paper shows that the same holds under the weaker assumption of finiteness of Davenport's constant of \(G_P\). The central point of the long combinatorial proof is an assertion about factorization patterns in factorization monoids of Abelian groups (Proposition 4.3), providing a bound on the cardinality of tame generating sets for pattern ideals.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 20M13
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 5549358
zbMATH Keywords: non-unique factorizations
zbMATH Keywords: lengths of factorizations
zbMATH Keywords: Krull monoids
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.11.020
OpenAlex ID: W2077048852
cites work: Q4674348
cites work: Q2718651
cites work: On minimal distances in Krull monoids with infinite class group
cites work: Krull Semigroups and Divisor Class Groups
cites work: Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids
cites work: Q3436763
cites work: Q3637545
cites work: Chains of Factorizations and Factorizations with Successive Lengths
cites work: An addition theorem and its arithmetical application
cites work: Zero-sum problems in finite Abelian groups: a survey
cites work: On products of \(k\) atoms.
cites work: Q5852785
cites work: Q3436766
cites work: Q5200682
cites work: Arithmetic of Mori domains and monoids
cites work: Local tameness of \(v\)-Noetherian monoids.
cites work: Q5584587
cites work: Non-unique factorizations of algebraic integers
cites work: Halbgruppen mit Divisorentheorie. (Semigroups with divisor theory)
cites work: Q4255424
cites work: Properties of factorizations with successive lengths in one-dimensional local domains
cites work: On Monoids of Finite Real Character
cites work: Weakly half-factorial sets in finite abelian groups
cites work: A realization theorem for sets of lengths
cites work: Direct-sum decompositions over local rings
DOI: 10.1016/J.JALGEBRA.2008.11.020
description: scientific article; zbMATH DE number 5549358