Ten Problems on Stability of Domains

Gabelli, Stefania

doi:10.1007/978-1-4939-0925-4_10

Cited by 6 publications

(4 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, we can say that in a stable Mori domain each divisorial ideal is 2-v-generated. Since a divisorial Mori domain is Noetherian, this result generalizes the fact that in a Noetherian domain that is stable and divisorial each ideal can be generated by two elements [49,Theorem 3.6].…”

Section: Problem 32a Is a Stable Archimedean Domain One-dimensional?mentioning

confidence: 62%

See 1 more Smart Citation

Open Problems in Commutative Ring Theory

Cahen¹,

Fontana

Frisch

et al. 2014

Commutative Algebra

View full text Add to dashboard Cite

This article consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of research approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were contributed by the authors and editors of this volume, as well as other researchers in the area.

show abstract

Section: Problem 32a Is a Stable Archimedean Domain One-dimensional?mentioning

confidence: 62%

“…The answer is positive in the semilocal case [54], so that a semilocal stable Archimedean domain is Mori (see also [49,Theorem 2.17]). Hence, a way of approaching this problem is trying to see if for stable domains the Archimedean property localizes.…”

Section: Problem 32a Is a Stable Archimedean Domain One-dimensional?mentioning

confidence: 99%

Open Problems in Commutative Ring Theory

Cahen¹,

Fontana

Frisch

et al. 2014

Commutative Algebra

View full text Add to dashboard Cite

show abstract

“…Bass rings arise naturally in geometry-as coordinate rings of (not necessarily irreducible) affine algebraic curves whose only singularities are double points, in number theory-for example, as quadratic orders, and in representation theory-in the form of Z[G] with G a finite abelian group of square-free order. Bass rings have Krull dimension at most one and belong to the larger class of stable rings -rings in which every ideal that contains a nonzerodivisor is projective over its ring of endomorphisms -see [25,44].…”

Section: Introductionmentioning

confidence: 99%

Lattices over Bass Rings and Graph Agglomerations

Baeth

Smertnig

2021

Algebr Represent Theor

View full text Add to dashboard Cite

We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T(R). Results of Levy–Wiegand and Levy–Odenthal together with a study of the local case yield an explicit description of T(R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations—a natural class of monoids serving as combinatorial models for the factorization theory of T(R). As a consequence, the monoid T(R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T(R) and characterize when T(R) is half-factorial. (Factoriality, that is, torsion-free Krull–Remak–Schmidt–Azumaya, is characterized by a theorem of Levy–Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest.

show abstract

Section: Introductionmentioning

confidence: 99%

Lattices over Bass rings and graph agglomerations

Baeth,

Smertnig

2020

Preprint

View full text Add to dashboard Cite

We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T (R). Results of Levy-Wiegand and Levy-Odenthal together with a study of the local case yield an explicit description of T (R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations-a natural class of monoids serving as combinatorial models for the factorization theory of T (R). As a consequence, the monoid T (R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T (R) and characterize when T (R) is half-factorial. (Factoriality, that is, torsionfree Krull-Remak-Schmidt-Azumaya, is characterized by a theorem of Levy-Odenthal.) The monoids of graph agglomerations introduced here may also be of independent interest to the factorization theory community.

show abstract

Ten Problems on Stability of Domains

Cited by 6 publications

References 38 publications

Open Problems in Commutative Ring Theory

Open Problems in Commutative Ring Theory

Lattices over Bass Rings and Graph Agglomerations

Lattices over Bass rings and graph agglomerations

Contact Info

Product

Resources

About