Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Chain conditions for epsilon-strongly graded rings with applications to Leavitt path algebras
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.ORCID iD: 0000-0001-8445-3936
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let $G$ be a group with neutral element $e$ and let $S=\bigoplus_{g \in G}S_g$ be a $G$-graded ring. A necessary condition for $S$ to be noetherian is that the principal component $S_e$ is noetherian. The following partial converse is well-known: If $S$ is strongly-graded and $G$ is a polycyclic-by-finite group, then $S_e$ being noetherian implies that $S$ is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings.

As our main application we obtain characterizations of noetherian and artinian Leavitt path algebras with coefficients in a general unital ring. This extends a recent characterization by Steinberg for Leavitt path algebras with coefficients in a commutative unital ring and previous characterizations by Abrams, Aranda Pino and Siles Molina for Leavitt path algebras with coefficients in a field. Secondly, we obtain characterizations of noetherian and artinian unital partial crossed products.

Keywords [en]
group graded ring, epsilon-strongly graded ring, chain conditions, Leavitt path algebra, partial crossed product.
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:bth-17807OAI: oai:DiVA.org:bth-17807DiVA, id: diva2:1304134
Funder
The Crafoord Foundation, 20170843Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2019-04-24Bibliographically approved
In thesis
1. The structure of epsilon-strongly graded rings with applications to Leavitt path algebras and Cuntz-Pimsner rings
Open this publication in new window or tab >>The structure of epsilon-strongly graded rings with applications to Leavitt path algebras and Cuntz-Pimsner rings
2019 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The research field of graded ring theory is a rich area of mathematics with many connections to e.g. the field of operator algebras. In the last 15 years, algebraists and operator algebraists have defined algebraic analogues of important operator algebras. Some of those analogues are rings that come equipped with a group grading. We want to reach a better understanding of the graded structure of those analogue rings. Among group graded rings, the strongly graded rings stand out as being especially well-behaved. The development of the general theory of strongly graded rings was initiated by Dade in the 1980s and since then numerous structural results have been established for strongly graded rings.

 In this thesis, we study the class of epsilon-strongly graded rings which was recently introduced by Nystedt, Öinert and Pinedo. This class is a natural generalization of the well-studied class of unital strongly graded rings. Our aim is to lay the foundation for a general theory of epsilon-strongly graded rings generalizing the theory of strongly graded rings. This thesis is based on three articles. The first two articles mainly concern structural properties of epsilon-strongly graded rings. In the first article, we investigate a functorial construction called the induced quotient group grading. In the second article, using results from the first article, we generalize the Hilbert Basis Theorem for strongly graded rings to epsilon-strongly graded rings and apply it to Leavitt path algebras.  In the third article, we study the graded structure of algebraic Cuntz-Pimsner rings. In particular, we obtain a partial classification of unital strongly, epsilon-strongly and nearly epsilon-strongly graded Cuntz-Pimsner rings up to graded isomorphism.

Place, publisher, year, edition, pages
Karlskrona: Blekinge Tekniska Högskola, 2019
Series
Blekinge Institute of Technology Licentiate Dissertation Series, ISSN 1650-2140 ; 7
Keywords
group graded ring, epsilon-strongly graded ring, chain conditions, Leavitt path algebra, partial crossed product, Cuntz-Pimsner rings
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-17809 (URN)978-91-7295-376-5 (ISBN)
Presentation
2019-05-15, G340, Valhallavägen 1, Karlskrona, 14:35 (English)
Opponent
Supervisors
Funder
The Crafoord Foundation, 20170843
Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2019-06-11Bibliographically approved

Open Access in DiVA

No full text in DiVA

Other links

https://arxiv.org/abs/1808.10163

Authority records BETA

Lännström, Daniel

Search in DiVA

By author/editor
Lännström, Daniel
By organisation
Department of Mathematics and Natural Sciences
Algebra and Logic

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 51 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf