Finiteness theorems for factorizations

Halter‐Koch, Franz

doi:10.1007/bf02574329

Cited by 38 publications

(24 citation statements)

References 7 publications

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“…Factorization theory can be studied at several levels of generality; much of it can be abstracted to semigroups, as in [8,9,11,12]. As our main interest is for integral domains, we will take the middle of the road and work with semigroups only when the added generality substantially simplifies the problem.…”

Section: Introductionmentioning

confidence: 99%

Rational elasticity of factorizations in Krull domains

Anderson¹,

Anderson²,

Chapman³

et al. 1993

Proc. Amer. Math. Soc.

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Abstract. For an atomic domain R , we define the elasticity of R as p(R) = sup{/n/« | xi •■■Xm -y\ •••>'«> for X,, ys e R irreducibles} and let Ir(x) and LR(x) denote, respectively, the inf and sup of the lengths of factorizations of a nonzero nonunit x € R into the product of irreducible elements. We answer affirmatively two rationality conjectures about factorizations. First, we show that p(R) is rational when R is a Krull domain with finite divisor class group. Secondly, we show that when R is a Krull domain, the two limits '«(■*")/" and Ln(x")/n , as n goes to infinity, are positive rational numbers. These answer, respectively, conjectures of D.

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Section: Introductionmentioning

confidence: 99%

Rational elasticity of factorizations in Krull domains

Anderson¹,

Anderson²,

Chapman³

et al. 1993

Proc. Amer. Math. Soc.

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“…They observed that an integral domain R has the property that R/I is finite for each proper (principal) ideal if and only if R is a field or R is a one-dimensional Noetherian domain with each residue field finite. [9]. This also follows from Theorem 2.…”

Section: Examplementioning

confidence: 52%

“…The next two theorems which generalize [9,Theorem 7] are straightforward modifications of its proof.…”

Section: Example 8 Let R ⊆ T Be a Pair Of Integral Domains And Letmentioning

confidence: 98%

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Finite factorization domains

Anderson

Mullins

1996

Proc. Amer. Math. Soc.

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Abstract. An integral domain R is a finite factorization domain if each nonzero element of R has only finitely many divisors, up to associates. We show that a Noetherian domain R is an FFD ⇔ for each overring R of R that is a finitely generated R-module, U (R )/U (R) is finite. For R local this is also equivalent to each R/[R : R ] being finite. We show that a one-dimensional local domain (R, M ) is an FFD ⇔ either R/M is finite or R is a DVR.In their study of factorization [2], the first author, D.F. Anderson, and M. Zafrullah introduced the notion of a finite factorization domain (FFD). An integral domain R is an FFD if every nonzero element of R has only a finite number of nonassociate divisors. The three authors continued their investigation of FFD's in [3], and F. Halter-Koch studied FFD's and their monoid analog in [9]. Earlier, A. Grams and H. Warner [8] introduced the related concept of idf-domains. An integral domain R is an idf-domain (for irreducible-divisor-finite) if each nonzero element of R has only finitely many nonassociate irreducible divisors.We adopt the following definitions and notation. For an integral domain R with quotient field K, U (R) is the group of units of R and G(R) = K * /U (R), partially ordered by aU (R) ≤ bU (R) ⇔ a|b in R, is the group of divisibility of R. Clearly G(R) is order-isomorphic to the group Prin(R) of nonzero principal fractional ideals of R ordered by reverse inclusion. We sometimes call an irreducible element of an integral domain an atom and an integral domain R is said to be atomic if every nonzero, nonunit element of R is a finite product of atoms. For an integral domain R, R * = R − {0} andR is the integral closure of R. For a survey of factorization in integral domains, the reader is referred to [2][3] and for standard definitions and results from commutative ring theory to [6] and [11].We begin by giving several equivalent conditions for an integral domain to be an FFD. Theorem 1. For an integral domain R, the following conditions are equivalent:(1) R is an FFD, (2) every nonzero (principal ) ideal of R is contained in only finitely many principal ideals, (3) for each x ∈ G(R) with x ≥ 0, the interval [0, x] is finite, (4) for any infinite collection of distinct principal ideals {(r α )} of R, α (r α ) = 0,

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“…It is well known that a monoid is a UFM if and only if it is an atomic AP-monoid. Following Anderson, Anderson, and Zafrullah [5] and Halter-Koch [29],…”

Section: Commutative Monoids and Factorizationsmentioning

confidence: 99%

On the primality and elasticity of algebraic valuations of cyclic free semirings

Jiang¹,

Li²,

Zhu³

2022

Preprint

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A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number α, the additive monoid M α of the evaluation semiring N 0 [α] is atomic. The atomic structure of both the additive and the multiplicative monoids of N 0 [α] has been the subject of several recent papers. Here we focus on the monoids M α , and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when α is less than 1, the atoms of N 0 [α] are as far from being prime as they can possibly be. Then we establish some results about the elasticity of N 0 [α], including that when α is rational, the elasticity of M α is full (this was previously conjectured by S. T. Chapman, F. Gotti, and M. Gotti).

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Finiteness theorems for factorizations

Cited by 38 publications

References 7 publications

Rational elasticity of factorizations in Krull domains

Rational elasticity of factorizations in Krull domains

Finite factorization domains

On the primality and elasticity of algebraic valuations of cyclic free semirings

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