Scott T. Chapman

Factorization in dedekind domains with finite class group

LetD be a Dedekind domain. It is well known thatD is then an atomic integral domain (that is to say, a domain in which each nonzero nonunit has a factorization as a product of irreducible elements). We study factorization properties of elements in Dedekind domains with finite class group. IfD has the property that any factorization of an elementα into irreducibles has the same length, thenD is called a half factorial domain (HFD, see [41]). IfD has the property that any factorization of an elementα into irreducibles has the same length modulor (for somer>1), thenD is called a congruence half factorial domain of orderr. In Section I we consider some general factorization properties of atomic integral domains as well as the interrelationship of the HFD and CHFD property in the Dedekind setting. In Section II we extend many of the results of [41], [42] and [36] concerning HFDs when the class group ofD is cyclic. Finally, in Section III we consider the CHFD property in detail and determine some basic properties of Dedekind CHFDs. IfG is any Abelian group andS any subset ofG−[0], then {G, S} is called a realizable pair if there exists a Dedekind domainD with class groupG such thatS is the set of nonprincipal classes ofG which contain prime ideals. We prove that for a finite abelian groupG there exists a realizable pair {G, S} such that any Dedekind domain associated to {G, S} is CHFD for somer>1 but not HFD if and only ifG is not isomorphic toZ2,Z2,Z2 ⊕Z2, orZ3 ⊕Z3.

Half-Factorial Domains, a Survey

Let D be an integral domain. D is atomic if every nonzero nonunit of D can be written as a product of irreducible elements (or atoms) of D. Let 1 (D) represent the set of irreducible elements of D. Traditionally, an atomic domain D is a unique factorization domain (UFD) if α 1… α n = β 1… β m for each ai and β j ∈I (D) implies: 1. n =m, 2. there exists a permutation б of {1,... ,n} such that α 1 and β б (i) are associates.

Minimal zero-sequences and the strong Davenport constant

Abstract Let G be a finite Abelian group and U (G) the set of minimal zero-sequences on G. If M 1 and M 2 ∈ U (G) , then set M 1 ∼ M 2 if there exists an automorphism ϕ of G such that ϕ( M 1 )= M 2 . Let O ( M ) represent the equivalence class of M under ∼. In this paper, we consider problems related to the size of an equivalence class of sequences in U (G) and also examine a stronger form of the Davenport constant of G.

Some factorization properties of Krull domains with infinite cyclic divisor class group

In this paper, we study factorization properties of Krull domains with divisor class group Z. This continues a preliminary study of Dedekind domains with class group Z in Section IV of [7]. In section 1, using the Φ-function we introduce the notion of a Φ-finite domain and then determine the relationship between these domains and BFDs and RBFDs (see [1]). In particular, we show that a Φ-finite domain need not be an RBFD. In Section 2, we obtain necessary and sufficient conditions on the set S of divisor classes of D which contain height-one prime ideals so that D is Φ-finite. This leads to the following result: if D is a Krull domain with divisor class group Z, then D is Φ-finite if and only if D is an RBFD. We also find a bound for the elasticity, ϱ(D), of the domain D and show in Section 3 that, unlike the case where the divisor class group of D is finite, the elasticity of D may not be “attained” by the factorization of a single element.

On the hfd, chfd, and k-hfd properties in dedekind domains

Let D be a Dedekind domain. D is a half factorial domain (HFD) if for any irreducible elements of D the equality implies that s = t. D is a congruence half factorial domain (CHFD) of order r>1 if the same equality implies that s = t (mod r). In this paper we expand upon many of the known results for HFDs and CHFDs (see [6] and [7]) as well as introduce the following new class of domains: if k≥1 is a positive integer then D is a k—ha1f factorial domain (k—HFD) if s Skin the previous equality implies that s = t. In section I we explore the interrelationship of the HFD, CHFD, and k—HFD properties and offer a method for constructing examples of k—HFDs and CHFDs by viewing the class group of the given domain as a direct summand. In particular, we show in section I that the HFD, CHFD, and k—HFD properties are equivalent for rings of algebraic integers. In section II we extend the results of section I by constructing examples of Dedekind domains which are both CHFD and k—HFD but not HFD. In section In we explore...

HOW FAR IS AN ELEMENT FROM BEING PRIME

Let D be an integral domain. We investigate two invariants ω(D, x) and ω(D) which measure how far an x ∈ D is from being prime and how far an atomic integral domain D is from being a UFD, respectively. We give a new characterization of number fields with class number two. We also study asymptotic versions of these two invariants.

Inside factorial monoids and integral domains

Abstract 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.

Monoids determined by a homogenous linear diophantine equation and the half-factorial property

Abstract We study additive submonoids M of N n consisting of the solutions of a homogeneous linear diophantine equation with integer coefficients. Surprisingly, not very much is known about the structure of M . M is a Krull monoid which, however, cannot be realized as a multiplicative monoid of a Krull domain. The concepts of divisor theory and divisor class group, nevertheless, do apply and we use them to characterize the factoriality of M in terms of the coefficients of the diophantine equation. In this paper, we concentrate on the more difficult question of finding conditions under which M is half-factorial. Since the famous Carlitz criterion of class number at most two breaks down for the Krull monoid M , we develop some new sufficient and/or necessary conditions for the half-factoriality of M . Among others, we present a geometric criterion for M to be half-factorial and an inequality condition on the coefficients of the diophantine equation assuring the half-factoriality of M .

Shifts of generators and delta sets of numerical monoids

Let S be a numerical monoid with minimal generating set 〈n1, …, nt〉. For m ∈ S, if

One Hundred Problems in Commutative Ring Theory

m = \sum_{i = 1}^{t} x_{i}n_{i}