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The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras

  • Dirceu Bagio ORCID logo , Daniel Gonçalves ORCID logo , Paula Savana Estácio Moreira ORCID logo EMAIL logo and Johan Öinert ORCID logo
Published/Copyright: January 3, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 36 Issue 4

Abstract

Given a partial action α of a groupoid G on a ring R, we study the associated partial skew groupoid ring R ⋊ α G , which carries a natural G-grading. We show that there is a one-to-one correspondence between the G-invariant ideals of R and the graded ideals of the G-graded ring R ⋊ α G . We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of R ⋊ α G . We show that every ideal of R ⋊ α G is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to G-simplicity of R, topological transitivity of θ is equivalent to G-primeness of R, and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra.

Keywords: Partial skew groupoid ring; residual intersection property; graded ideal; topological freeness; topological transitivity; condition (K); ultragraph algebra
MSC 2020: 16S35; 16S99; 20L05; 16N60; 16W22; 16W50; 16S88

Communicated by Siegfried Echterhoff

Funding source: Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina

Award Identifier / Grant number: 001

Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

Award Identifier / Grant number: 307910/2020-2

Funding source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Award Identifier / Grant number: 001

Funding statement: The second named author was partially supported by Fapesc – Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina, Capes-Print (Finance Code 001), and CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico – Brazil (Finance Code 307910/2020-2). The third named author was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.

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Received: 2023-04-04
Revised: 2023-10-04
Published Online: 2024-01-03
Published in Print: 2024-07-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The C*-algebra of the Boidol group
  3. Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order
  4. Positive rigs
  5. Torus bundles over lens spaces
  6. Topological amenability of semihypergroups
  7. On projections of the tails of a power
  8. Li–Yorke chaos for composition operators on Orlicz spaces
  9. A note on the post quantum-Sheffer polynomial sequences
  10. Finite rigid sets of the non-separating curve complex
  11. Building planar polygon spaces from the projective braid arrangement
  12. Octonionic monogenic and slice monogenic Hardy and Bergman spaces
  13. Transcendence on algebraic groups
  14. An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes
  15. The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras
  16. Joint distribution of the cokernels of random p-adic matrices II
https://doi.org/10.1515/forum-2023-0117
Keywords for this article
Partial skew groupoid ring; residual intersection property; graded ideal; topological freeness; topological transitivity; condition (K); ultragraph algebra

Articles in the same Issue

  1. Frontmatter
  2. The C*-algebra of the Boidol group
  3. Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order
  4. Positive rigs
  5. Torus bundles over lens spaces
  6. Topological amenability of semihypergroups
  7. On projections of the tails of a power
  8. Li–Yorke chaos for composition operators on Orlicz spaces
  9. A note on the post quantum-Sheffer polynomial sequences
  10. Finite rigid sets of the non-separating curve complex
  11. Building planar polygon spaces from the projective braid arrangement
  12. Octonionic monogenic and slice monogenic Hardy and Bergman spaces
  13. Transcendence on algebraic groups
  14. An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes
  15. The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras
  16. Joint distribution of the cokernels of random p-adic matrices II
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