On minimal distances in Krull monoids with infinite class group

Communicated by E.M. Friedlander MSC: 13A05; 13F05; 20M13 a b s t r a c tLet H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b ∈ H, let ω(H, b) be the smallest N ∈ N 0 ∪ {∞} having the following property: if n ∈ N and a 1 , . . . , a n ∈ H are such that b divides a 1 · . . . · a n , then b already divides a subproduct of a 1 · . . . · a n consisting of at most N factors. The monoid H is called tame if sup{ω(H, u) | u is an atom of H} < ∞. This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.

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“…The set ∆ * (G) has been investigated in detail (see [35,10,36]), and in general it is a proper subset of ∆(H). …”

Section: Remark 52 We Say That the Structure Theorem For Sets Of Lementioning

confidence: 99%

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Geroldinger

Kainrath

2010

Journal of Pure and Applied Algebra

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“…The assumption that every class of G contains a prime divisor has a strong impact on the factorizations in H. Unions of sets of lengths are intervals [8, 10, Theorem 3.1.3], and if the set of classes of prime divisors of some element a ∈ H form a subgroup of G, then the set of lengths L(a) is an interval [12, Theorem 7.6.9]. The structure of a crucial subset of Δ(H) was studied in [5].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The set of distances in Krull monoids

Geroldinger

Yuan

2012

Bulletin of the London Mathematical Society

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Let H be a Krull monoid with class group G and suppose that every class contains a prime divisor. The set of distances Δ(H) of H is the set of all d ∈ N with the following property: there are irreducible elements u 1, . . . , u k , v1, . . . , v k+d such that u1 · . . . · u k = v1 · . . . · v k+d , but u 1 · . . . · u k cannot be written as a product of l irreducible elements for any l with k < l < k + d. We show that Δ(H) is an interval.

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“…The set Δ * (G) has found considerable interest (see [14,Sections 4.3 and 6.8], [3,23]), and in general, Δ * (G) is a proper subset of Δ(H). If G P is infinite, then the occurring differences of the AAMPs are actually in Δ(H) and not in the subset Δ * (G P ) (see [14,Example 4.8.10]).…”

Section: Moreover the Bound M Depends Only On D(g P ) If D(g P ) 3 mentioning

confidence: 99%

“…In order to give an explicit bound, note that it suffices to do this for B(G P ). Suppose that D(G P ) 3 …”

Section: Moreover the Bound M Depends Only On D(g P ) If D(g P ) 3 mentioning

confidence: 99%

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On the arithmetic of Krull monoids with finite Davenport constant

Geroldinger

Grynkiewicz

2009

Journal of Algebra

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Let H be a Krull monoid with class group G, G P ⊂ G the set of classes containing prime divisors and D(G P ) the Davenport constant of G P . We show that the finiteness of the Davenport constant implies the Structure Theorem for Sets of Lengths. More precisely, if D(G P ) < ∞, then there exists a constant M-for which we derive an explicit upper bound in terms of D(G P )-such that the set of lengths of every element a ∈ H is an almost arithmetical multiprogression with bound M.

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On minimal distances in Krull monoids with infinite class group

Abstract: Let H be a Krull monoid with infinite class group such that each divisor class contains a prime divisor. It is shown that for every positive integer n, there exists a divisor closed submonoid S of H such that min Δ(S) = n.

Cited by 15 publications

References 15 publications

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

The set of distances in Krull monoids

On the arithmetic of Krull monoids with finite Davenport constant

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