Monoids of modules and arithmetic of direct-sum decompositions

Baeth, Nicholas R.; Geroldinger, Alfred

doi:10.2140/pjm.2014.271.257

Cited by 42 publications

(76 citation statements)

References 67 publications

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“…For an overview we refer to the monograph of Leuschke and Wiegand [29]. We highlight a result of particular relevance to us by Baeth and Geroldinger [1,Theorem 5.5], yielding Krull monoids with finite cyclic class group (of any order) such that each class contains a prime divisor (earlier example often had infinite class groups).…”

Section: Monoids and Factorizationsmentioning

confidence: 99%

On congruence half-factorial Krull monoids with cyclic class group

Plagne

Schmid

2019

J. Comb. Algebra

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We carry out a detailed investigation of congruence half-factorial Krull monoids with finite cyclic class group and related problems. Specifically, we determine precisely all relatively large values that can occur as a minimal distance of a Krull monoid with finite cyclic class group, as well as the exact distribution of prime divisors over the ideal classes in these cases. Our results apply to various classical objects, including maximal orders and certain semi-groups of modules. In addition, we present applications to quantitative problems in factorization theory. More specifically, we determine exponents in the asymptotic formulas for the number of algebraic integers whose sets of lengths have a large difference.

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Section: Monoids and Factorizationsmentioning

confidence: 99%

On congruence half-factorial Krull monoids with cyclic class group

Plagne

Schmid

2019

J. Comb. Algebra

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“…Monoid domains and power series domains that are Krull are discussed in [2,19], and note that every class of a Krull monoid domain contains a prime divisor. For monoids of modules that are Krull and their distribution of prime divisors, we refer the reader to [1,5].…”

Section: Completely Integrally Closed and V-noetherianmentioning

confidence: 99%

Sets of minimal distances and characterizations of class groups of Krull monoids

Zhong

2017

Ramanujan J

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Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then every non-unit a ∈ H can be written as a finite product of atoms, say a = u 1 · . . . · u k . The set L(a) of all possible factorization lengths k is called the set of lengths of a. There is a constant M ∈ N such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ * (H ), where * (H ) denotes the set of minimal distances of H . We study the structure of * (H ) and establish a characterization when * (H ) is an interval. The system L(H ) = {L(a) | a ∈ H } of all sets of lengths depends only on the class group G, and a standing conjecture states that conversely the system L(H ) is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to C r n with r, n ∈ N and * (H ) is not an interval.

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“…We just mention that a domain R is a Krull domain if and only if its monoid of nonzero elements is a Krull monoid, and for monoids of modules which are Krull we refer to [2,1,9]. Property (a) easily shows that every integrally closed noetherian domain is a Krull domain.…”

Section: Preliminariesmentioning

confidence: 99%

“…Their suprema ρ k (H) = sup U k (H) were first studied in the 1980s for rings of integers in algebraic number fields ( [8,19]). Since then these invariants have been studied in a variety of settings, including numerical monoids, monoids of modules, noetherian and Krull domains (for a sample out of many we refer to [10,4,3,15,1]). …”

Section: Introductionmentioning

confidence: 99%