Half-factorial Domain Research Articles

Elasticity in Polynomial-Type Extensions

AbstractThe elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a half-factorial domain. We consider the relationship between the elasticity of a domain R and the elasticity of its polynomial ring R[x]. For example, if R has at least one atom, a sufficient condition for the polynomial ring R[x] to have elasticity 1 is that every non-constant irreducible polynomial f ∈ R[x] be irreducible in K[x]. We will determine the integral domains R whose polynomial rings satisfy this condition.

Half-factorial domains and quadratic orders

In this study, we investigate the half-factorial property in non-maximal real quadratic orders and investigate some necessary and sufficient conditions for these orders to be half-factorial domains in terms of their conductor.

Half-factorial subrings of factorial domains

A half-factorial domain D is a domain in which every non-zero element that is not a unit is a product of a unique number of irreducible elements of D. We characterize half-factorial subrings R of factorial domains S when S is the integral closure of R and their unit groups are identical. Let A be a factorial domain and A[T] the polynomial ring over A in the variable T. The characterization is used to describe the half-factorial A-subalgebras R with multiplicative conductors of A[T] into R.

Half-factoriality in subrings of trigonometric polynomial rings

Trigonometric polynomials are widely used in different fields of engineering and science. Inspired by their applications, we investigate half-factorial domains in trigonometric polynomial rings. We construct the half-factorial domains T '2, T'3 and T'4 which are the subrings of the ring of complex trigonometric polynomials T , such that T'2 ⊆ T'3 ⊆ T'4 ⊆ T'. We also discuss among these three subrings the Condition: Let A ⊆ B be a unitary (commutative) ring extension. For each ∈ B there exist ∈ U (B) and x ∈ A such that = x'x, where U (B) denote the group of units of B.

A quasi-local half-factorial domain with an atomic non-half-factorial integral closure

A quasi-local half-factorial domain with an atomic non-half-factorial integral closure

Every divisor class of Krull monoid domains contains a prime ideal

Let D be an integral domain, Γ be a torsion-free grading monoid, and D [ Γ ] be the monoid domain of Γ over D. Suppose that D [ Γ ] is a Krull domain, and let Cl ( D [ Γ ] ) be the divisor class group of D [ Γ ] . We show that every divisor class of D [ Γ ] contains a prime ideal. As a corollary, we have that D [ Γ ] is a half-factorial domain if and only if | Cl ( D [ Γ ] ) | ⩽ 2 ; hence in this case, either D or Γ is factorial. We also show that if T is the set of non-homogeneous prime elements of D [ Γ ] , then D [ Γ ] T is a π-domain with Cl ( D [ Γ ] ) = Cl ( D [ Γ ] T ) .

On unique factorization domains

In this paper we attempt to generalize the notion of “unique factorization domain” in the spirit of “half-factorial domain”. It is shown that this new generalization of UFD implies the now well-known notion of half-factorial domain. As a consequence, we discover that one of the standard axioms for unique factorization domains is slightly redundant.

Factorization in Integral Domains, IV

Let D be an integral domain such that every nonzero nonunit of D is a finite product of irreducible elements. In this article, we introduce and study several unifying concepts for the theory of nonunique factorization in D. They give a new way to measure, in some sense, how far an half-factorial domain (resp., bounded factorization domain, atomic domain) D is from being a UFD (resp., finite factorization domain, Cohen–Kaplansky domain) based on equivalence relations on the set of irreducible elements of D.

Factorization in Integral Domains without Identity

We investigate factorization in a nonzero integral domain A without identity. Now A may be viewed as a proper ideal of an integral domain D with identity having the same quotient field as A where \(A[1] \subseteq D \subseteq A:A\). We study the relationship between factorization in A and D, especially in the case where D = A[1] or A:A. Particular interest is given to various notions of unique factorization in A and to the relationship between A or D being atomic, satisfying ACCP, or being a bounded factorization domain, half-factorial domain, or finite factorization domain.

Power series extensions of half-factorial domains

An integral domain is said to be a half-factorial domain (HFD) if every non-zero element a that is not a unit may be factored into a finite product of irreducible elements, while any other such factorization of a has the same number of irreducible factors. While it is known that a power series extension of a factorial domain need not be factorial, the corresponding question for HFD has been open. In this paper we show that the answer is also negative. In the process we answer in the negative, for HFD, an open question of Samuel for factorial domains by showing that for certain quadratic domains R , and independent variables, Y and T , R [ [ Y ] ] [ [ T ] ] is not HFD even when R [ [ Y ] ] is HFD. The proof hinges on Samuel’s theorem to the effect that a power series, in finitely many variables, over a regular factorial domain is factorial.

A NOTE ON ASCEND AND DESCEND OF FACTORIZATION PROPERTIES

In this paper we extend the study of ascend and descend of factorization properties (for atomic domains, domains satisfying ACCP, bounded factorization domains, half-factorial domains, pre-Schreier and semirigid domains) to the finite factorization domains and idf-domains for domain extension <TEX>$A\;{\subseteq}\;B$</TEX>.

Boundary valuation domains

A half factorial domain (HFD) R is an atomic domain where, for any collection of irreducibles { α 1, α 2,…, α m , β 1, β 2,…, β n }, with α 1 α 2⋯ α m = β 1 β 2⋯ β n we have n= m. In a paper by J. Coykendall [Comm. Algebra 27 (1999) 3153–3159], a generalization of the length function of Zaks [Israel J. Math. 37 (1980) 281–302], called the boundary map, was introduced. A new class of HFD's—called boundary valuation domains—are defined and studied using this map.

On the integral closure of a half-factorial domain

A half-factorial domain (HFD), R, is an atomic integral domain where given any two products of irreducible elements of R: α 1α 2⋯α n=β 1β 2⋯β m then n= m. As a natural generalization of unique factorization domains (UFD), one wishes to investigate which “good” properties of UFDs that HFDs possess. In particular, it has been conjectured that the integral closure of a half-factorial domain is again a HFD (see Non-Noetherian Commutative Ring Theory, Mathematics and its applications, Vol. 520, Kluwer, Dordrecht, 2000, pp. 97–115. for example). In this paper we produce an example that demonstrates that the integral closure of a HFD does not even have to be atomic. We shall investigate the failure of this conjecture closely and highlight some cases where the conjecture does indeed hold.

Remarks on pseudo-valuation rings

A prime ideal P of a ring A is said to be a strongly prime ideal if aP and bA are comparable for all a,b ϵ A. We shall say that a ring A is a pseudo-valuation ring (PVR) if each prime ideal of A is a strongly prime ideal. We show that if A is a PVR with maximal ideal M, then every overring of A is a PVR if and only if M is a maximal ideal of every overring of M that does not contain the reciprocal’of any element of M.We show that if R is an atomic domain and a PVD, then dim(R) ≤ 1. We show that if R is a PVD and a prime ideal of R is finitely generated, then every overring of R is a PVD. We give a characterization of an atomic PVD in terms of the concept of half-factorial domain.

A Factorization Formula for Class Number Two

Let R be an atomic integral domain. R is a half-factorial domain (HFD) if whenever x1…xn= y1…ym for x1, …, xn, y1, …, ym irreducibles of R, then n=m. A well known result of L. Carlitz (1960, Proc. Amer. Math. Soc.11, 391–392) states that the ring of integers in a finite extension of the rationals is a HFD if and only if the class number of R is less than or equal to 2. If R is such a ring of integers with class number 2, then we use some simple Krull monoids to develop a formula for counting the number of different factorizations of any integer x into products of irreducible elements of R.

Elasticity of A + XB[ X] domains

Let A ⊂ B be an extension of integral domains and X be an indeterminate over B. In this paper, we study the elasticity of atomic domains of the form A + XB[ X]. We pay particular attention to when such domains are half-factorial domains, and in particular, we answer a question raised by V. Barucci et al. Finally, we investigate the elasticity of Z[√d] + X Z[ (1 + √d) 2 ][X] , where d is a nonzero square-free integer, d  1 ( mod 4). This provides an example such that A + XB[ X] is a HFD where A and B are HFDs but not UFDs.

Overring of half-factorial domains. II

An atomic integral domain D is a half-factorial domain (HFD) if for any irreducible elements α1,…,αn,β1,…,βmof D with α1,…,αn= β1,…,βm, then n = m. We explore some general properties of an integral domain D for which each localization of D is a HFD. In [5], we constructed an example of a Dedekind domain with divisor class group Z 6 which is a HFD, but with a localization which is not a HFD. We show that this construction can be extended to the case where the divisor class group of D is any finite abelian group except 1)cyclic p-group, and 2)direct sums of copies of Z 2. We close with a look at the relationship between the elasticity of an atomic domain and the elasticity of its localization.

Overrings of Half-Factorial Domains

AbstractAn atomic integral domain D is a half-factorial domain (HFD) if for any irreducible elements α1,..., αn, β1,..., βm of D with α1... αn = β1 ...βm, then n = m. In [3], Anderson, Anderson, and Zafrullah explore factorization problems in overrings of HFDs and ask whether a localization of a HFD is again a HFD. We construct an example of a Dedekind domain which is a HFD, but with a localization which is not a HFD. We also give an example of a Dedekind domain where each localization is a HFD, but with an overring which is not a HFD.

On the k ‐HFD property in Dedekind domains with small class group

Let D be an atomic integral domain (i.e., a domain in which each nonzero nonunit of D can be written as a product of irreducible elements) and k any positive integer. D is known as a half factorial domain (HFD) if for any irreducible elements α1, …, αn, β1, …, βm of D the equality α1… αn = β1… βm implies that n = m. In [5] the present authors define D to be a k-half factorial domain (k-HFD) if the equality above along with the fact that n or m ≤ k implies that n = m. In this paper we consider the k-HFD property in Dedekind domains with small class group and prove the following Theorem: if D is a Dedekind domain with class group of order less than 16 then D is k-HFD for some integer k > 1, if, and only if, D is HFD.

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...