On minimal distances in Krull monoids with infinite class group

2008 ◽  
Vol 40 (4) ◽  
pp. 613-618 ◽  
Author(s):  
S. T. Chapman ◽  
W. A. Schmid ◽  
W. W. Smith
Keyword(s):  
Class Group ◽  
Infinite Class ◽  
Krull Monoids

scholarly journals Factorization in Krull monoids with infinite class group

1999 ◽  
Vol 80 (1) ◽  
pp. 23-30 ◽  
Author(s):  
Florian Kainrath
Keyword(s):  
Class Group ◽  
Infinite Class ◽  
Krull Monoids

scholarly journals Factorization properties of Krull monoids with infinite class group

2002 ◽  
Vol 92 (2) ◽  
pp. 229-242 ◽  
Author(s):  
Wolfgang Hassler
Keyword(s):  
Class Group ◽  
Infinite Class ◽  
Krull Monoids

scholarly journals Differences in sets of lengths of Krull monoids with finite class group

2005 ◽  
Vol 17 (1) ◽  
pp. 323-345 ◽  
Author(s):  
Wolfgang A. Schmid
Keyword(s):  
Class Group ◽  
Krull Monoids ◽  
Sets Of Lengths

HIGHER-ORDER CLASS GROUPS AND BLOCK MONOIDS OF KRULL MONOIDS WITH TORSION CLASS GROUP

2010 ◽  
Vol 09 (03) ◽  
pp. 433-464 ◽  
Author(s):  
WOLFGANG A. SCHMID
Keyword(s):  
Class Group ◽  
Point Of View ◽  
Torsion Class ◽  
Class Groups ◽  
Detailed Understanding ◽  
Krull Monoids ◽  
Krull Monoid ◽  
Zero Sum ◽  
Block Monoids ◽  
Block Monoid

Extensions of the notion of a class group and a block monoid associated to a Krull monoid with torsion class group are introduced and investigated. Instead of assigning to a Krull monoid only one abelian group (the class group) and one monoid of zero-sum sequences (the block monoid), we assign to it a recursively defined family of abelian groups, the first being the class group, and do alike for the block monoid. These investigations are motivated by the aim of gaining a more detailed understanding of the arithmetic of Krull monoids, including Dedekind and Krull domains, both from a technical and conceptual point of view. To illustrate our method, some first arithmetical applications are presented.

A NEW CHARACTERIZATION OF HALF-FACTORIAL KRULL MONOIDS

2010 ◽  
Vol 09 (05) ◽  
pp. 825-837 ◽  
Author(s):  
PAUL BAGINSKI ◽  
ROSS KRAVITZ
Keyword(s):  
Class Group ◽  
Finite Product ◽  
Krull Monoids ◽  
Irreducible Elements ◽  
Krull Monoid ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

Let M be a Krull monoid. Then every element of M may be written as a finite product of irreducible elements. If for every a ∈ M, each two factorizations of a have the same number of irreducible elements, then M is called half-factorial. Using a property of element exponentiation, we provide a new characterization of half-factoriality, valid for all Krull monoids whose class group has torsion-free rank at most one.

scholarly journals THE CATENARY DEGREE OF KRULL MONOIDS II

2014 ◽  
Vol 98 (3) ◽  
pp. 324-354 ◽  
Author(s):  
ALFRED GEROLDINGER ◽  
QINGHAI ZHONG
Keyword(s):  
Class Group ◽  
Algebraic Function ◽  
Algebraic Number Field ◽  
Class Groups ◽  
Davenport Constant ◽  
Algebraic Function Field ◽  
Krull Monoids ◽  
Krull Monoid ◽  
Catenary Degree

Let$H$be a Krull monoid with finite class group$G$such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree$\mathsf{c}(H)$of$H$is the smallest integer$N$with the following property: for each$a\in H$and each pair of factorizations$z,z^{\prime }$of$a$, there exist factorizations$z=z_{0},\dots ,z_{k}=z^{\prime }$of$a$such that, for each$i\in [1,k]$,$z_{i}$arises from$z_{i-1}$by replacing at most$N$atoms from$z_{i-1}$by at most$N$new atoms. To exclude trivial cases, suppose that$|G|\geq 3$. Then the catenary degree depends only on the class group$G$and we have$\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where$\mathsf{D}(G)$denotes the Davenport constant of$G$. The cases when$\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldingeret al.[‘The catenary degree of Krull monoids I’,J. Théor. Nombres Bordeaux23(2011), 137–169], we determine the class groups satisfying$\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of$\mathsf{c}(H)$is known for no further class groups.

scholarly journals On transfer homomorphisms of Krull monoids

2021 ◽  
Author(s):  
Alfred Geroldinger ◽  
Florian Kainrath
Keyword(s):  
Class Group ◽  
Finitely Generated ◽  
Krull Monoids ◽  
Krull Monoid ◽  
Transfer Homomorphism ◽  
Zero Sum

AbstractEvery Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group.

scholarly journals On the arithmetic of Krull monoids with infinite cyclic class group

2010 ◽  
Vol 214 (12) ◽  
pp. 2219-2250 ◽  
Author(s):  
A. Geroldinger ◽  
D.J. Grynkiewicz ◽  
G.J. Schaeffer ◽  
W.A. Schmid
Keyword(s):  
Class Group ◽  
Krull Monoids

scholarly journals On congruence half-factorial Krull monoids with cyclic class group

2019 ◽  
Vol 3 (4) ◽  
pp. 331-400
Author(s):  
Alain Plagne ◽  
Wolfgang Schmid
Keyword(s):  
Class Group ◽  
Krull Monoids

An inequality concerning the elasticity of Krull monoids with divisor class group $\mathbb{Z}_p$

2007 ◽  
Vol 56 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Scott T. Chapman ◽  
William W. Smith
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Krull Monoids
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