Arithmetic of Mori Domains and Monoids: The Global Case

Kainrath, Florian

doi:10.1007/978-3-319-38855-7_8

Cited by 10 publications

(4 citation statements)

References 11 publications

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“…In other words, a v-noetherian monoid is not necessarily locally tame. Apart from Krull monoids which will be discussed below, main examples of locally tame monoids are C-monoids: if R is a noetherian domain with integral closure R, non-zero conductor f, finite residue field R/f and finite class group C(R), then R is a C-monoid, and there is an explicit characterization when R is globally tame (see [12,Theorem 2.11.9] and [13,18,25]).…”

Section: Tameness and Transfer Homomorphismsmentioning

confidence: 99%

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Local and Global Tameness in Krull Monoids

Gao

Geroldinger

Schmid

2014

Communications in Algebra

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Section: Tameness and Transfer Homomorphismsmentioning

confidence: 99%

“…The concepts of local and global tameness have found some attention in recent literature, and they were studied in settings ranging from numerical monoids to noetherian domains (confer [5,14,4,15,11,23,24,2,21,18,22]). We recall their definitions.…”

Section: Introductionmentioning

confidence: 99%

Local and Global Tameness in Krull Monoids

Gao

Geroldinger

Schmid

2014

Communications in Algebra

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“…In Section 4 we will characterize when the monoid V((C) is tame for C a class of finitely generated modules over a commutative Noetherian local ring (Proposition 4.1). Further examples of tame monoids can be found in [30,39]. We now gather together several arithmetical finiteness properties of tame monoids.…”

Section: Preliminariesmentioning

confidence: 99%

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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“…If H is not half-factorial, then a simple argument shows that min ∆(H) = gcd ∆(H) ( [10,Proposition 1.4.4]). Sets of distances are finite for transfer Krull monoids of finite type (hence in particular for Krull domains with finite class group), weakly Krull domains with finite v-class group, finitely generated monoids, and others (see [9,Theorem 13], [10, Theorems 3.1.4 and 3.7.1], [13]). The question which finite sets can actually occur as a set of distances (of any monoid or domain) was open so far.…”

Section: Introductionmentioning

confidence: 99%