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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
##### 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
OAI: 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.

##### 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)
##### Funder
The Crafoord Foundation, 20170843 Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2019-06-11Bibliographically approved

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#### Authority records BETA

Lännström, Daniel

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##### By author/editor
Lännström, Daniel
##### By organisation
Department of Mathematics and Natural Sciences
##### On the subject
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Cite
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