Big indecomposable modules and direct-sum relations

Häßler, Wolfgang; Karr, Ryan; Klingler, Lee; Wiegand, Roger

doi:10.1215/ijm/1258735327

Cited by 9 publications

(6 citation statements)

References 17 publications

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“…The past decade has seen a new semigroup-theoretical approach. This approach was first introduced by Facchini and Wiegand [29] and has been used by several authors (for example, see [3], [4], [7], [8], [19], [22], [25], [26], [27], [28], [29], [46], [49], and [57]). Let R be a ring and let C be a class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Monoids of modules and arithmetic of direct-sum decompositions

Baeth

Geroldinger

2014

Pacific J. Math.

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“…The notion of a Krull monoid also covers other monoids of interest. The monoids formed by the isomorphy classes of certain modules under directsum decomposition are Krull monoids (see, for example, [8,16]). Moreover, the monoids of zero-sum sequences over subsets of abelian groups are Krull monoids, and conversely, many questions regarding factorizations in Krull monoids can be transferred to questions regarding zero-sum sequences over subsets of their class groups (see Section 2 for details).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On minimal distances in Krull monoids with infinite class group

Chapman

Schmid²,

Smith³

2008

Bulletin of the London Mathematical Society

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“…Our first lemma summarizes some basic information about C. We first consider the first statement of 3. The class group of V(C) is free abelian of finite rank by [38,Theorem 6.3]. Therefore, by statement 2, the class group G of V(C ′ ) is finitely generated.…”

Section: Monoids Of Modules Over Commutative Noetherian Local Ringsmentioning

confidence: 95%

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

et al. 2014

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Abstract. Let R be a ring and let C be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let V(C) denote a set of representatives of isomorphism classes in C and, for any module M in C, let [M ] denote the unique element in V(C) isomorphic to M . Then V(C) is a reduced commutative semigroup with operation defined by [M ] , and this semigroup carries all information about direct-sum decompositions of modules in C. This semigroup-theoretical point of view has been prevalent in the theory of direct-sum decompositions since it was shown that if End R (M ) is semilocal for all M ∈ C, then V(C) is a Krull monoid. Suppose that the monoid V(C) is Krull with a finitely generated class group (for example, when C is the class of finitely generated torsion-free modules and R is a one-dimensional reduced Noetherian local ring). In this case we study the arithmetic of V(C) using new methods from zero-sum theory. Furthermore, based on module-theoretic work of Lam, Levy, Robson, and others we study the algebraic and arithmetic structure of the monoid V(C) for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

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Big indecomposable modules and direct-sum relations

Cited by 9 publications

References 17 publications

Monoids of modules and arithmetic of direct-sum decompositions

Monoids of modules and arithmetic of direct-sum decompositions

On minimal distances in Krull monoids with infinite class group

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

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