Irreducible Divisor Graphs and Factorization Properties of Domains

This article examines the connections between the factorization properties of a domain, e.g., unique factorization domain (UFD), finite factorization domain (FFD), and the domain's irreducible divisor graphs. In particular, we show that although there are some nice correlations between the properties of the domain D and the set of irreducible divisor graphs {G(x): x ∈ D* \ U(D)} when D is an FFD, it is very unlikely that any information about the domain D can be gleaned from the collection {G(x): x ∈ D* \ U(D)} when D is not an FFD. We also introduce an alternate irreducible divisor graph called the compressed irreducible divisor graph and study some of its properties.     

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 toZ ₂,Z ₂,Z ₂ ⊕Z ₂, orZ ₃ ⊕Z ₃. The first author received support under the John M. Bennett Fellowship at Trinity University and also gratefully acknowledges the support of The University of North Carolina at Chapel Hill.     

On almost-divided domains

Let B be a domain, Q a maximal ideal of B, π: B → B/Q the canonical surjection, D a subring of B/Q, and A:=π ⁻¹(D). If both B and D are almost-divided domains (resp., n-divided domains), then A = B × _B/Q D is an almost-divided domain (resp., an n-divided domain); the converse holds if B is quasilocal. If 2 ≤ d ≤ ∞, an example is given of an almost-divided domain of Krull dimension d which is not a divided domain.      

FACTORIZATION IN MONOID DOMAINS

In this paper, we determine necessary and sufficient conditions for the group ring D[G] to be a BFD (resp., an FFD, an SFFD). Also we give necessary and sufficient conditions for the monoid domain D[S] to be a BFD (resp., an FFD, an SFFD). In addition, we characterize when themonoid domain D[S] is a UFD in terms of 2-factoriality.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

GCD-Sets in Integral Domains. II

Abstract

Let D be an integral domain and S ≠ U(D) a saturated multiplicative subset of D. We say that S is a GCD-set (resp., factorial-set) if S is a GCD-monoid (resp., factorial-monoid) under the product of D and that S is a Marot-set if every integral ideal of D intersecting S is generated by a set of elements in S. In this paper, we study Marot GCD-sets and Marot factorial-sets.     

唯一分解整环R上不可约多项式的一个判别准则

本获得了一个判别唯一分解整环R及其商域Q上的n次（n＞2)多项式不可约的充分条件。     

Ascending Nets of Prüfer V-Multiplication Domains

Abstract

Let D be an integral domain, S ? D a multiplicative set such that aD _S  ∩ D is a principal ideal for each a ∈ D and let D ^(S) = ? _s∈S D[X/s]. It is known that if D is a Prüfer v-multiplication domain (resp., generalized GCD domain, GCD domain), then so is D ^(S) respectively. When D is a Noetherian domain, we obtain a similar result for the power series analog D ^((S)) = ? _s∈S D[[X/s]] of D ^(S). Our approach takes care simultaneously of both cases D ^(S) and D ^((S)).     

唯一分解整环上多项式不可约的一个判别法

给出了唯一分解整环上多项式不可约的一个判别法.     

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?₀, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X ⁿ K[X] = {a + X ⁿ g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X ⁿ K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X ⁿ K[X].     

一类无界域的Szegö核

本文研究了由任意不可约有界齐次圆域构造的一类无界域D_ψ的Szegö核.利用Cartan域上一类积分的明显表达式,获得了无界域D_ψ的Szegö核的明显公式.     

Finiteness and Skolem Closure of Ideals for Non-Unibranched Domains

A one-dimensional, Noetherian, local domain D with maximal ideal 𝔪 and finite residue field was known to be an almost strong Skolem ring if analytically irreducible. It was unknown whether this condition is necessary. We show that it is at least necessary for D to be unibranched. After introducing a general notion of equalizing ideal, we show that, for k large enough, the ideals of the form 𝔐 _{k, a}  = {f ∈ Int(D) | f(a) ∈ 𝔪 ^k }, for a ∈ D, are distinct. This allows to show that the maximal ideals 𝔐 _a  = {f ∈ Int(D) | f(a) ∈ 𝔪}, although not necessarily distinct, are never finitely generated.     

Supersoluble and Related Cofinitary Groups

Let D be a division ring (possibly commutative) and V an infinite-dimensional left vector space over D. We consider irreducible subgroups G of GL(V) containing an element whose fixed-point set in V is non-zero but finite dimensional (over D). We then derive conclusions about cofinitary groups, an element of GL(V) being cofinitary if its fixed-point set is finite dimensional and a subgroup G of GL(V) being cofinitary if all its non-identity elements are confinitary. In particular we show that an irreducible cofinitary subgroup G of GL(V) is usually imprimitive if G is supersoluble and is frequently imprimitive if G is hypercyclic. The latter includes the case where G is hypercentral, which apparently is also new.     

有限域上n元n次方程只有零解的充要条件

本文研究了有限域上只有零解的n元n次方程的结构问题.利用对有限域上不可约多元多项式在其扩域中的分解特征的刻画,结合Chevalley定理,得到了有限域上n元n次方程只有零解的一个充要条件,并给出这类方程的一些新的具体构造.     

Uppers to Zero in Polynomial Rings and Prüfer-Like Domains

Let D be an integral domain and X an indeterminate over D. It is well known that (a) D is quasi-Prüfer (i.e., its integral closure is a Prüfer domain) if and only if each upper to zero Q in D[X] contains a polynomial g ∈ D[X] with content c _D (g) = D; (b) an upper to zero Q in D[X] is a maximal t-ideal if and only if Q contains a nonzero polynomial g ∈ D[X] with c _D (g) ^v  = D. Using these facts, the notions of UMt-domain (i.e., an integral domain such that each upper to zero is a maximal t-ideal) and quasi-Prüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this article, given a semistar operation ☆ in the sense of Okabe–Matsuda, we introduce the ☆-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UMt-domains and the Prüfer v-multiplication domains.     

Factorizations and characterizations of induced‐hereditary and compositive properties

An Erratum has been published for this article in Journal of Graph Theory 50:261, 2005 . A graph property (i.e., a set of graphs) is hereditary (respectively, induced‐hereditary) if it is closed under taking subgraphs (resp., induced‐subgraphs), while the property is additive if it is closed under disjoint unions. If and are properties, the product consists of all graphs G for which there is a partition of the vertex set of G into (possibly empty) subsets A and B with G[A] and G[B] . A property is reducible if it is the product of two other properties, and irreducible otherwise. We show that very few reducible induced‐hereditary properties have a unique factorization into irreducibles, and we describe them completely. On the other hand, we give a new and simpler proof that additive hereditary properties have a unique factorization into irreducible additive hereditary properties [J. Graph Theory 33 (2000), 44–53]. We also introduce analogs of additive induced‐hereditary properties, and characterize them in the style of Scheinerman [Discrete Math. 55 (1985), 185–193]. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 11–27, 2005     

Elasticity in certain block monoids via the Euclidean table

This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤ _n , S) where S = (hereafter denoted ℬ _a (n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬ _a (n). In Section 3 we use the Euclidean table to compute the elasticity of ℬ _a (n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬ _a (n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬ _a (p). Part of this work was completed while the second author was on an Academic Leave granted by the Trinity University Faculty Development Committee.     

1. Introduction

In this paper, we give the conception of implicit congruence and nonimplicit congruence in a unique factorization domain R and establish some structures of irreducible polynomials over R. A classical result, Eisenstein's criterion, is generalized.     

MCD-finite Domains and Ascent of IDF Property in Polynomial Extensions

An integral domain is said to have the IDF property when every non-zero element of it has only a finite number of non-associate irreducible divisors. A counterexample has already been found showing that the IDF property does not necessarily ascend in polynomial extensions. In this paper, we introduce a new class of integral domains, called MCD-finite domains, and show that for any domain D, D[X] is an IDF domain if and only if D is both IDF and MCD-finite. This result entails all the previously known sufficient conditions for the ascent of the IDF property. Our new characterization of polynomial domains with the IDF property enables us to use a different construction and build another counterexample which strengthen the previously known result on this matter.