Krull Monoids Research Articles

A NEW CHARACTERIZATION OF HALF-FACTORIAL KRULL MONOIDS

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.

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

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.

On the arithmetic of Krull monoids with infinite cyclic class group

Let H be a Krull monoid with infinite cyclic class group G and let G P ⊂ G denote the set of classes containing prime divisors. We study under which conditions on G P some of the main finiteness properties of factorization theory–such as local tameness, the finiteness and rationality of the elasticity, the structure theorem for sets of lengths, the finiteness of the catenary degree, and the existence of monotone and near monotone chains of factorizations–hold in H . In many cases, we derive explicit characterizations.

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b ∈ H , let ω ( H , b ) be the smallest N ∈ N 0 ∪ { ∞ } having the following property: if n ∈ N and a 1 , … , a n ∈ H are such that b divides a 1 ⋅ … ⋅ a n , then b already divides a subproduct of a 1 ⋅ … ⋅ a n consisting of at most N factors. The monoid H is called tame if sup { ω ( H , u ) ∣ u is an atom of H } < ∞ . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M .

On the Delta set and catenary degree of Krull monoids with infinite cyclic divisor class group

Let M be a Krull monoid with divisor class group Z , and let S ⊆ Z denote the set of divisor classes of M which contain prime divisors. We find conditions on S equivalent to the finiteness of both Δ ( M ) , the Delta set of M , and c ( M ) , the catenary degree of M . In the finite case, we obtain explicit upper bounds on max Δ ( M ) and c ( M ) . Our methods generalize and complement a previous result concerning the elasticity of M .

A realization theorem for sets of lengths

Text By a result of G. Freiman and A. Geroldinger [G. Freiman, A. Geroldinger, An addition theorem and its arithmetical application, J. Number Theory 85 (1) (2000) 59–73] it is known that the set of lengths of factorizations of an algebraic integer (in the ring of integers of an algebraic number field), or more generally of an element of a Krull monoid with finite class group, has a certain structure: it is an almost arithmetical multiprogression for whose difference and bound only finitely many values are possible, and these depend just on the class group. We establish a sort of converse to this result, showing that for each choice of finitely many differences and of a bound there exists some number field such that each almost arithmetical multiprogression with one of these difference and that bound is up to shift the set of lengths of an algebraic integer of that number field. Moreover, we give an explicit sufficient condition on the class group of the number field for this to happen. Video For a video summary of this paper, please visit http://www.youtube.com/watch?v=c61xM-5D6Do.

Arithmetical characterization of class groups of the form ℤ/nℤ⊕ℤ/nℤ via the system of sets of lengths

Let H be a Krull monoid with finite class group such that each class contains a prime divisor (e.g., the multiplicative monoid of the ring of algebraic integers of some number field). It is shown that it can be determined whether the class group is of the form ℤ/nℤ⊕ℤ/nℤ, for n≥3, just by considering the system of sets of lengths of H.

On the arithmetic of Krull monoids with finite Davenport constant

Let H be a Krull monoid with class group G, G P ⊂ G the set of classes containing prime divisors and D ( G P ) the Davenport constant of G P . We show that the finiteness of the Davenport constant implies the Structure Theorem for Sets of Lengths. More precisely, if D ( G P ) < ∞ , then there exists a constant M—for which we derive an explicit upper bound in terms of D ( G P ) —such that the set of lengths of every element a ∈ H is an almost arithmetical multiprogression with bound M.

On products of k atoms

Let H be an atomic monoid. For \(k \in {\Bbb N}\) let \({\cal V}_k (H)\) denote the set of all \(m \in {\Bbb N}\) with the following property: There exist atoms (irreducible elements) u 1, …, u k , v 1, …, v m ∈ H with u 1· … · u k = v 1 · … · v m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets \({\cal V}_k (H)\) are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every \(k \in {\Bbb N}\), max \({\cal V}_{2k+1} (H) = k \vert G\vert + 1\) which settles Problem 38 in [4].

On minimal distances in Krull monoids with infinite class group

Let H be a Krull monoid with infinite class group such that each divisor class contains a prime divisor. It is shown that for every positive integer n, there exists a divisor closed submonoid S of H such that min Δ(S) = n.

Some Characterizations of Krull Monoids

In this paper, Kaplansky-type theorems are given to characterize GCD-monoids and valuation monoids. Also, (unique) r-factorable monoids are defined and it is shown that S is a Krull monoid if and only if S is a unique t-factorable (resp., w-factorable) monoid if and only if S is a t-factorable (resp., w-factorable) t-Prüfer monoid.

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

Let p be a prime integer and M a Krull monoid with divisor class group $\mathbb{Z}_p$ . We represent by S the set of nontrivial divisor classes of $\mathbb{Z}_p$ which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best possible. If M as above has | S| = 3, then it is known that $\rho(M)\in \{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}$ , but for large p, not all the values in this containment set can be realized. For each | S| = 3, we produce a submonoid $\widetilde{\mathcal{B}}(\mathbb{Z}_p,S)$ of $\mathcal{B}(\mathbb{Z}_p,S)$ such that $$ \{\,\rho(\widetilde{\mathcal{B}}(\mathbb{Z}_p,S))\,\mid \, S\subseteq \mathbb{Z}_p-\{0\}\mbox{ and }\mid S\mid =3\}=\{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}. $$ Keywords: Krull monoid, Block monoid, Elasticity of factorization Mathematics Subject Classification (2000): 20M14, 20D60, 13F05

Geometric regularity of direct-sum decompositions in some classes of modules

In this paper, we show that modules with semilocal endomorphism rings appear in abundance in applications, that their direct-sum decompositions are described by the so-called Krull monoids, and that this implies a geometric regularity of the direct-sum decompositions of these modules. Their direct-sum decompositions into indecomposables are not necessarily unique in the sense of the Krull-Schmidt theorem. The application of the theory of Krull monoids to the study of direct-sum decompositions of modules has been developed during the last five years. After a quick survey of the results obtained in this direction, we concentrate in particular on the abundance of examples. At present, these examples are scattered in the literature, and we try to collect them in a systematic way.

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

Soit H un monoïde de Krull dont le groupe de classes est fini. On suppose que chaque classe contient un diviseur premier. On sait que tout ensemble de longueurs est une presque multiprogression arithmétique. Nous étudions les nombres entiers qui apparaissent comme raison de ces progressions. Nous obtenons en particulier une borne supérieure sur la taille de ces raisons. En appliquant ces résultats, nous pouvons montrer que, sauf dans un cas particulier connu, deux p-groupes élémentaires ont le même système d’ensembles de longueurs si et seulement si ils sont isomorphes.

On The Asymptotic Values of Length Functions In Krull And Finitely Generated commutative Monoids

AbstractLet M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.

ON FACTORIZATION IN BLOCK MONOIDS FORMED BY $\{\bar{1},\bar{a}\}$ IN $\mathbb{Z}_{n}$

AbstractWe consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determine the elasticity of $\mathcal{B}_a(n)$ based solely on the cross number. Section 4 applies the results of §3 to investigate the complete set of elasticities of Krull monoids with divisor class group $\mathbb{Z}_n$.AMS 2000 Mathematics subject classification: Primary 20M14; 20D60; 13F05

Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids

Commutative monoids yield an analogy between the theory of factorization in commutative integral domains and the theory of direct sum decompositions of modules. We show that the monoid V( C ) of isomorphism classes of a class C of modules with semilocal endomorphism rings is a Krull monoid (Theorem 3.4). Krull monoids often appear in the study of factorizations of elements in integral domains, and are defined as the monoids V for which there is a divisor homomorphism of V into a free commutative monoid. In particular, we consider the case in which C is the class of biuniform modules. For this class the validity of a weak form of the Krull–Schmidt Theorem is explained via a representation of V( C ) as a subdirect product of free commutative monoids.

On the number of factorizations of an element in an atomic monoid

Let S be a reduced commutative cancellative atomic monoid. If s is a nonzero element of S, then we explore problems related to the computation of η(s), which represents the number of distinct irreducible factorizations of s∈S. In particular, if S is a saturated submonoid of Nd, then we provide an algorithm for computing the positive integer r(s) for which 0<limn→∞η(sn)nr(s)−1<∞. We further show that r(s) is constant on the Archimedean components of S. We apply the algorithm to show how to compute limn→∞η(sn)nr(s)−1 and also consider various stability conditions studied earlier for Krull monoids with finite divisor class group.

Inside factorial monoids and integral domains

We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisor-theoretic manner to the concept of unique factorization. We prove that a monoid is outside factorial if and only if it is a Krull monoid with torsion class group, and that it is inside factorial if and only if its root-closure is a rational generalized Krull monoid with torsion class group. We determine the structure of Cale bases of inside factorial monoids and characterize inside factorial monoids among weakly Krull monoids. These characterizations carry over to integral domains. Inside factorial orders in algebraic number fields are characterized by several other factorization properties.

Factorization properties of Krull monoids with infinite class group

{For a non-unit $a$ of an atomic monoid $H$ we call $$L_H(a) = \{k\in {\mathbb N} \mid a = u_1 \ldots u_k \hbox{ with irreducible } u_i\in H\}$$ the set of lengths of $a$. Let $H$ be a Krull monoid with infinite divisor class group such that each divisor