Structural Additive Theory

Grynkiewicz, David J.

doi:10.1007/978-3-319-00416-7

Cited by 140 publications

(135 citation statements)

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“…The study of sequences, subsequence sums, and zero-sums is a flourishing subfield of Additive Group and Number Theory (see, for example, [31], [40], and [43]). The Davenport constant D(G 0 ), defined as…”

Section: Krull Monoids Monoids Of Modules and Transfer Homomorphismsmentioning

confidence: 99%

Monoids of modules and arithmetic of direct-sum decompositions

Baeth

Geroldinger

2014

Pacific J. Math.

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Abstract. Let R be a (possibly noncommutative) ring and let C be a class of finitely generated (right) R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set V(C) of isomorphism classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup encodes all possible information about direct sum decompositions of modules in C. If the endomorphism ring of each module in C is semilocal, then V(C) is a Krull monoid. Although this fact was observed nearly a decade ago, the focus of study thus far has been on ring-and module-theoretic conditions enforcing that V(C) is Krull. If V(C) is Krull, its arithmetic depends only on the class group of V(C) and the set of classes containing prime divisors. In this paper we provide the first systematic treatment to study the direct-sum decompositions of modules using methods from Factorization Theory of Krull monoids. We do this when C is the class of finitely generated torsion-free modules over certain one-and two-dimensional commutative Noetherian local rings.

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Section: Krull Monoids Monoids Of Modules and Transfer Homomorphismsmentioning

confidence: 99%

Monoids of modules and arithmetic of direct-sum decompositions

Baeth

Geroldinger

2014

Pacific J. Math.

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“…In particular, zero-sum theoretical invariants (such as the Davenport constant or the cross number) and the associated inverse problems play a crucial role (surveys and detailed presentations of such results can be found in [8,10,17]). Most of these invariants are 722 Q. Zhong well understood only in a very limited number of cases (e.g., for groups of rank two, the precise value of the Davenport constant D(G) is known and the associated inverse problem is solved; however, if n is not a prime power and r ≥ 3, then the precise value of the Davenport constant D(C r n ) is unknown).…”

Section: Given Two Finite Abelian Groups G and G Withmentioning

confidence: 99%

“…Our notation and terminology are consistent with [8,10,17]. Let N denote the set of positive integers and N 0 = N ∪ {0}.…”

Section: Background On Krull Monoids and Their Sets Of Minimal Distancesmentioning

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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“…By a sequence over G, we mean a finite sequence of terms from G where repetition is allowed and the order is disregarded. As usual (see [14,17]), we consider sequences as elements of the free abelian monoid F (G) with basis G. A sequence S over G will be written in the form…”

Section: Preliminariesmentioning

confidence: 99%

“…Based on the Savchev-Chen Structure Theorem [17,Section 11.3] (resp. on a related result on the index of sequences) Gao and Geroldinger [11] showed that for every cyclic group G and every k ∈ N we have ρ 2k+1 (H) = kD(G) + 1.…”

Section: Introductionmentioning

confidence: 99%