Product Of Irreducible Elements Research Articles

A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

ABSTRACTLet H be a transfer Krull monoid over a finite abelian group G (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit a∈H can be written as a product of irreducible elements, say , and the number of factors k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system ℒ(H) = {L(a)∣a∈H} of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system ℒ(H) is characteristic for the group G. Let H′ be a further transfer Krull monoid over a finite abelian group G′ and suppose that ℒ(H) = ℒ(H′). We prove that, if with r≤n−3 or (r≥n−1≥2 and n is a prime power), then G and G′ are isomorphic.

Factorizations of upper triangular Toeplitz matrices

Let D be a principal ideal domain and m a positive integer. We denote by \(R_m(D)\) the ring of \((m+1) \times (m + 1)\) upper triangular Toeplitz matrices with entries in D. Then \(R_m(D)\cong R[X]/(X^{m+1})\) is a commutative Noetherian ring with identity and thus every nonzero non-unit of \(R_m(D)\) can be written as a finite product of irreducible elements. In this manuscript we give conditions for an element of \(R_m(D)\) to be irreducible, describe the general structure of factorizations of elements as products of irreducible elements, and give a description of how certain elements factor as products of irreducible elements. In particular, we generalize work of Chang and Smertnig in the case where \(m=1\).

Factorizations of upper triangular matrices

If D is an integral domain in which every element can be factored as a product of irreducible elements, then every element A∈Tn(D), the semigroup of upper triangular matrices with nonzero determinant, can be factored as a product of irreducible elements. We classify both the irreducible elements and units of this cancellative semigroup. Having achieved this, we provide a means of measuring the uniqueness or non-uniqueness of these factorizations in terms of the uniqueness or non-uniqueness of factorizations in D. In addition to using well-studied invariants from factorization theory of commutative semigroups, we introduce a new tool, weak transfer homomorphisms, which are more appropriate for noncommutative settings.

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.

On the asymptotic behavior of unions of sets of lengths in atomic monoids

Let M be a commutative cancellative atomic monoid. We use unions of sets of lengths in M to construct the V-Delta set of M . We first derive some basic properties of V-Delta sets and then show how they offer a method to investigate the asymptotic behavior of the sizes of unions of sets of lengths. A central focus of number theory is the study of number theoretic functions and their asymptotic behavior. This has led to similar investigations concerning non-unique factorizations in integral domains and moniods. Suppose that M is a commutative cancellative monoid in which each nonunit can be factored into a product of irreducible elements (such a monoid is known as atomic). For a nonunit x in M , let L(x) represent the maximum length of a factorization of x into irreducibles and l(x) the minimum such length. The functions L(x) = lim k→∞ L(xn) n and l(x) = lim k→∞ l(xn)

Asymptotic Elasticity in Atomic Monoids

Let M be a commutative atomic monoid (i.e. every nonzero nonunit of M can be factored as a product of irreducible elements). Let ρ(x) denote the elasticity of x ∈ M, R(M) = {ρ(x) | x ∈ M} the set of elasticities of elements in M, and ρ(M) = sup R(M) the elasticity of M. Define \overline{ρ}(x) = limn→∞ ρ(xn) to be the asymptotic elasticity of x. We determine some basic properties of the function \overline{ρ} and determine its image for certain block monoids.

Chains of Factorizations and Factorizations with Successive Lengths

ABSTRACT Let H be a commutative cancellative monoid. H is called atomic if every nonunit a ∈ H decomposes (in general in a highly nonunique way) into a product of irreducible elements (atoms) u i of H. The integer n is called the length of (†), and L(a) = {n ∈ ℕ | a decomposes into n irreducible elements of H} is called the set of lengths of a. Two integers k < l are called successive lengths of a if L(a) ∩ {m ∈ ℕ | k ≤ m ≤ l} = {k, l}. For a ∈ H we denote by Z n (a) the set of factorizations of a with length n. Suppose now that H is one of the following monoids: i. A congruence monoid in a Dedekind domain with finite residue fields. ii. H = D\\ {0}, where D is a Noetherian domain having the following properties: is a Krull domain with finite divisor class group, and is a finite ring. iii. H = D\\ {0}, where D is a one-dimensional Noetherian domain with finite normalization and finite Picard group (but possibly infinite residue fields). Let a ∈ H. In the present article, we investigate the structure of concatenating chains in Z n (a) as well as the relation between Z k (a) and Z l (a) if k and l are successive lengths of a. The work continues earlier investigations in Foroutan (2003), Foroutan and Geroldinger (2004), and Hassler (to appear).

Polynomial extensions of atomic domains

Among other results, we obtain here a normal atomic domain A such that A[ X] is not atomic but any reducible element in A is a product of two irreducible elements (an atomic domain is a domain in which every nonzero nonunit is a product of irreducible elements).

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.

Length Functions on Integral Domains

Let $R$ be an integral domain and $x \in R$ which is a product of irreducible elements. Let $l(x)$ and $L(x)$ denote respectively the inf and sup of the lengths of factorizations of $x$ into a product of irreducible elements. We show that the two limits, $\bar l(x)$ and $\bar L(x)$, of $l({x^n})/n$ and $L({x^n})/n$, respectively, as $n$ goes to infinity always exist. Moreover, for any $0 \leq \alpha \leq 1 \leq \beta \leq \infty$, there is an integral domain $R$ and an irreducible $x \in R$ such that $\bar l(x) = \alpha$ and $\overline L (x) = \beta$.

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.

Factorization in integral domains

In this paper, we study factorization in an integral domain R, that is, factoring elements of R into products of irreducible elements. We investigate several factorization properties in R which are weaker than unique factorization.

Left perfect rings that are right perfect and a characterization of Steinitz rings

A proof is given to show all flat left modules of a ring are free if and only if the ring is a local ring with a left Tnilpotent maximal ideal. We characterize left perfect rings whose radical R has the property that IRn={O} for some positive integer n if 1 is a finitely generated right ideal contained in R. We cite an example of a left perfect ring which does not have this property. It is shown that if the set of irreducible elements of a left perfect ring is right T-nilpotent then the ring is right perfect. Introduction. If R is the radical of a left perfect ring, an element of R is called irreducible if it cannot be expressed as a product of two elements of R. If R is the radical of a ring that is left and right perfect, we show each element of R can be expressed as a product of irreducible elements. It follows that a perfect ring having a finite set of irreducible elements is right and left artinian and has a finite radical; furthermore if the ring is a local ring with a nonzero radical, the ring is finite. Throughout this paper we will assume that a ring has an identity element and that a module is unitary. DEFINITION. Let F be a free left A-module having {uj}1=1 as a basis; let {aj}n1 be a sequence in A and let G denote the submodule of F generated by {Uj-ajuj+1} 1. The pair Fand G will be denoted [F, {aj}, G]. LEMMA 1. FIG is a flat left A-module. PROOF. It is sufficient to show that if I is a right ideal, IFnG=IG. If mI Secondary 16A46.