Atomic Integral Domain Research Articles

On the ascent of atomicity to monoid algebras

A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid M and an integral domain R are both atomic implies that the monoid algebra R[M] of M over R is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, we prove here that the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors.

Atoms in quasilocal integral domains and Cohen–Kaplansky domains

Let [Formula: see text] be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of [Formula: see text]. Special attention is given to the set of atoms in [Formula: see text] and to the existence of atoms in [Formula: see text]. While our main interest is in local Cohen–Kaplansky (CK) domains (atomic integral domains with only finitely many nonassociate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight non-associate atoms having an atom in [Formula: see text].

On the atomicity of monoid algebras

Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whether the monoid ring R[x;M] is atomic provided that both M and R are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for M=N0: he constructed an atomic integral domain R such that the polynomial ring R[x] is not atomic. However, the question of whether a monoid algebra F[x;M] over a field F is atomic provided that M is atomic has been open since then. Here we offer a negative answer to this question. First, we exhibit for any infinite cardinal κ a torsion-free atomic monoid M of rank κ satisfying that the monoid domain R[x;M] is not atomic for any integral domain R. Then for every n≥2 and for each field F of finite characteristic we find a torsion-free atomic monoid M of rank n such that F[x;M] is not atomic. Finally, we construct a torsion-free atomic monoid M of rank 1 such that Z2[x;M] is not atomic.

Proper divisibility in computable rings

We study divisibility in computable integral domains. We develop a technique for coding Σ20 binary trees into the divisibility relation of a computable integral domain. We then use this technique to prove two theorems about non-atomic integral domains.In every atomic integral domain, the divisibility relation is well-founded. We show that this classical theorem is equivalent to ACA0 over RCA0.In every computable non-atomic integral domain there is a Δ30 infinite sequence of proper divisions. We show that this upper bound cannot be improved to Δ20 in general.

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.

ON BOUNDING MEASURES OF PRIMENESS IN INTEGRAL DOMAINS

Let D be an integral domain. In this paper, we investigate two (integer- or ∞-valued) invariants ω(D, x) and ω(D) which measure how far a nonzero x ∈ D is from being prime and how far an atomic integral domain D is from being a unique factorization domain (UFD), respectively. In particular, we are interested in when there is a nonzero (irreducible) x ∈ D with ω(D, x) = ∞ and the relationship between ω(A, x) and ω(B, x), and ω(A) and ω(B), for an extension A ⊆ B of integral domains and a nonzero x ∈ A.

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.

Full Elasticity in Atomic Monoids and Integral Domains

Full Elasticity in Atomic Monoids and Integral Domains

The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids

Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S)) and the tame degree of S (denoted t(S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe methods to compute both c(S) and t(S) when M is a finitely generated commutative cancellative monoid.

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.

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.

Integral Domains with Highly Nonunique Factorization

For a positive integer n, an atomic integral domain R is defined to be completely non- n- factorial if for any n atoms a1…, an, the product a1… an has as highly nonunique a factorization into atoms as possible in that given any n − 1 atoms b1,…, bnt - 1, b1 … bn− 1¦a1… an. We show that R is completely non-n-factorial for some n ≥ 2 if and only if (R, M) is a quasilocal domain with [M: M] a DVR having M as its maximal ideal.

Factorization Sets and Half-Factorial Sets in Integral Domains

Let R be an atomic integral domain. Suppose that H is a nonempty subset of irreducible elements of R, u is a unit of R, and α 1,..., α n , β 1,..., β m are irreducible elements of R such that (1) α 1... α n = u · β 1... β m . Set H α = { i|α i ∈ H } and H β = { j|β j ∈ H }. H is a factorization set ( F-set) of R if for any equality involving irreducibles of the form (1), | H α| ≠ 0 implies that | H β| ≠ 0. H is a half-factorial set ( HF-set) if any equality of the form (1) implies that | H α| = | H β|. In this paper, we explore in detail the structure of the F-sets and HF-sets of an atomic integral domain R. If P is a nonzero prime ideal of R and J ( R) the set of irreducible elements of R, then set H P = P ∩ J ( R). We show that if F is an F-set of R, then there exists a nonempty set X of nonzero prime ideals of R such that F = ∪ P ∈ X H P . We define an F-set to be minimal if it contains no proper subsets which are F-sets. We then show that in R every F-set can be written as a union of minimal F-sets if and only if R satisfies the Principal Ideal Theorem. If, in addition, every height-one prime ideal of R is the radical of a principal ideal, then this representation is unique. We study the structure of the HF-sets of R and concentrate on the case where R is a Krull domain with torsion divisor class group. For such a domain R, we show that if H is an HF-set of R and P is a nonprincipal prime ideal of R with H P ⊆ H , then H Q ⊆ H for each prime ideal Q of R in the same divisor class as P. We also show that if R is a Dedekind domain with prime class number p ≥ 2 and P ( R) is the set of prime elements of R, then H an HF-set of R implies that either H ∪ P ( R) = J ( R) or H ⊆ P ( R).

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.

Elasticity of factorizations in atomic monoids and integral domains

Elasticity of factorizations in atomic monoids and integral domains

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.

Factorization in LCM Domains with Conjugation

AbstractAn atomic integral domain with conjugation has unique (in the sense of Theorem 6 below) factorization of atomic factors if it is an LCM domain. If the LCM hypothesis is dropped not even the number of atomic factors in a complete factorization of an element need be unique.

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.

Elasticity of factorizations in integral domains

For an atomic integral domain R, define ϱ( R)= sup{ m⧸ n| x 1⋯ x m = y 1⋯ y n , each x i , y j ϵ R is irreducible}. We investigate ϱ( R), with emphasis for Krull domains R. When R is a Krull domain, we determine lower and upper bounds for ϱ( R); in particular, ϱ( R)≤ max{| Cl( R)|⧸ 2, 1}. Moreover, we show that for any real numbers r≥1 or r=∞, there is a Dedekind domain R with torsion class group such that ϱ( R)= r.

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.