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.

SELF-SMALL ABELIAN GROUPS

2009 ◽  
Vol 80 (2) ◽  
pp. 205-216 ◽  
Author(s):  
ULRICH ALBRECHT ◽  
SIMION BREAZ ◽  
WILLIAM WICKLESS
Keyword(s):  
Small Groups ◽  
Abelian Groups ◽  
Direct Sums ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractThis paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.

Nilpotent-by-Noetherian Factorized Groups

1989 ◽  
Vol 32 (4) ◽  
pp. 391-403
Author(s):  
Bernhard Amberg ◽  
Silvana Franciosi ◽  
Francesco de Giovanni
Keyword(s):  
Factor Group ◽  
Finite Product ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractIt is shown that a soluble-by-finite product G = AB of a nilpotent-by-noetherian group A and a noetherian group B is nilpotentby- noetherian. Moreover, a bound for the torsion-free rank of the Fitting factor group of G is given, in terms of the torsion-free rank of the Fitting factor group of A and the torsion-free rank of B.

scholarly journals Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

2017 ◽  
Vol 16 (12) ◽  
pp. 1750234 ◽  
Author(s):  
Yushuang Fan ◽  
Alfred Geroldinger ◽  
Florian Kainrath ◽  
Salvatore Tringali
Keyword(s):  
Structure Theorem ◽  
Unit Element ◽  
Finite Product ◽  
New Approach ◽  
Irreducible Elements ◽  
Abstract Approach ◽  
Krull Monoid ◽  
Arithmetic Combinatorics ◽  
First Time

Let [Formula: see text] be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every [Formula: see text], let [Formula: see text] denote the set of all [Formula: see text] with the property that there are atoms [Formula: see text] such that [Formula: see text] (thus, [Formula: see text] is the union of all sets of lengths containing [Formula: see text]). The Structure Theorem for Unions states that, for all sufficiently large [Formula: see text], the sets [Formula: see text] are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

On the Square Subgroup of a Mixed SI-Group

2018 ◽  
Vol 61 (1) ◽  
pp. 295-304 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
M. Woronowicz
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Negative Solution ◽  
New Class ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Group Modulo

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

scholarly journals The torsion-free rank of homology in towers of soluble pro-p groups

2017 ◽  
Vol 219 (2) ◽  
pp. 817-834 ◽  
Author(s):  
Martin R. Bridson ◽  
Dessislava H. Kochloukova
Keyword(s):  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

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.

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