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The structure of epsilon-strongly graded rings with applications to Leavitt path algebras and Cuntz-Pimsner rings
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.ORCID iD: 0000-0001-8445-3936
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 [en]
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: urn:nbn:se:bth-17809ISBN: 978-91-7295-376-5 (print)OAI: oai:DiVA.org:bth-17809DiVA, id: diva2:1304154
Presentation
2019-05-15, G340, Valhallavägen 1, Karlskrona, 14:35 (English)
Opponent
Supervisors
Funder
The Crafoord Foundation, 20170843Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2019-06-11Bibliographically approved
List of papers
1. Induced quotient group gradings of epsilon-strongly graded rings
Open this publication in new window or tab >>Induced quotient group gradings of epsilon-strongly graded rings
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let $G$ be a group and let $S=\bigoplus_{g \in G} S_g$ be a $G$-graded ring. Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading of $S$. It is well-known that if $S$ is strongly $G$-graded, then the induced $G/N$-grading is strong for any $N$. The class of epsilon-strongly graded rings was recently introduced by Nystedt, Öinert and Pinedo as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and sufficient conditions for the induced $G/N$-grading of an epsilon-strongly $G$-graded ring to be epsilon-strong. Our method involves relating different types of rings equipped with local units (s-unital rings, rings with sets of local units, rings with enough idempotents) with generalized epsilon-strongly graded rings. 

Keywords
group graded ring, epsilon-strongly graded ring, Leavitt path algebra, partial skew group ring.
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-17806 (URN)
Funder
The Crafoord Foundation, 20170843
Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2019-04-24Bibliographically approved
2. Chain conditions for epsilon-strongly graded rings with applications to Leavitt path algebras
Open this publication in new window or tab >>Chain conditions for epsilon-strongly graded rings with applications to Leavitt path algebras
(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
group graded ring, epsilon-strongly graded ring, chain conditions, Leavitt path algebra, partial crossed product.
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-17807 (URN)
Funder
The Crafoord Foundation, 20170843
Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2019-04-24Bibliographically approved
3. The graded structure of algebraic Cuntz-Pimsner rings
Open this publication in new window or tab >>The graded structure of algebraic Cuntz-Pimsner rings
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We  classify strongly, epsilon-strongly and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. As an application, we characterize noetherian and artinian fractional skew monoid rings by a single corner automorphism.

Keywords
group graded ring, epsilon-strongly graded ring, Cuntz-Pimsner ring, Leavitt path algebra, fractional skew monoid ring
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-17808 (URN)
Funder
The Crafoord Foundation, 20170843
Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2019-04-24Bibliographically approved

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