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The structure of epsilon-strongly group graded rings
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.ORCID iD: 0000-0001-8445-3936
2021 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The development of a general theory of strongly group graded rings was initiated by Dade, Năstăsescu and Van Oystaeyen in the 1980s, and since then numerous structural results have been established.  In this thesis we develop a general theory of so-called (nearly) epsilon-strongly group graded rings which were recently introduced by Nystedt, Öinert and Pinedo and which generalize strongly group graded rings. Moreover, we obtain applications to  Leavitt path algebras, unital partial crossed products and algebraic Cuntz-Pimsner rings. 

This thesis is based on five scientific papers (A, B, C, D, E). 

Papers A and B are concerned with structural properties of epsilon-strongly graded rings. In Paper A, we consider an important construction called the induced quotient group grading. In Paper B, using results from Paper A, we obtain a Hilbert Basis Theorem for epsilon-strongly graded rings.  In Paper C, we study the graded structure of algebraic  Cuntz-Pimsner rings. In particular, we obtain a partial characterization of unital strongly graded, epsilon-strongly graded and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. 

In Paper D, we give a complete characterization of group graded rings that are graded von Neumann regular.

Finally, in Paper E, written in collaboration with Lundström, Öinert and Wagner, we consider prime nearly epsilon-strongly graded rings. Generalizing Passman's work from the 1980s, we give  necessary and sufficient conditions for a nearly epsilon-strongly graded ring to be prime. 

Place, publisher, year, edition, pages
Karlskrona: Blekinge Tekniska Högskola, 2021.
Series
Blekinge Institute of Technology Doctoral Dissertation Series, ISSN 1653-2090 ; 3
Keywords [en]
group graded ring, Leavitt path algebra, partial crossedproduct, Cuntz-Pimsner ring, von Neumann regular ring, non-unital ring
National Category
Algebra and Logic
Research subject
Mathematics and applications
Identifiers
URN: urn:nbn:se:bth-21342ISBN: 978-91-7295-421-2 (print)OAI: oai:DiVA.org:bth-21342DiVA, id: diva2:1546251
Public defence
2021-09-01, Zoom/J1630, 15:00 (English)
Opponent
Supervisors
Available from: 2021-04-21 Created: 2021-04-21 Last updated: 2021-06-14Bibliographically approved
List of papers
1. 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
2020 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 23, no 4, p. 1707-1726Article in journal (Refereed) Published
Abstract [en]

Let G be a group with neutral element e and let S=⊕g∈GSg be a G-graded ring. A necessary condition for S to be noetherian is that the principal component Se is noetherian. The following partial converse is well-known: If S is strongly-graded and G is a polycyclic-by-finite group, then Se 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. © 2019, The Author(s).

Place, publisher, year, edition, pages
Springer Netherlands, 2020
Keywords
Chain conditions, Epsilon-strongly graded ring, Group graded ring, Leavitt path algebra, Partial crossed product, Mathematical techniques, Chain condition, Finite groups, Neutral elements, Path algebras, Principal Components, Algebra
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-17807 (URN)10.1007/s10468-019-09909-0 (DOI)000550240900022 ()
Funder
The Crafoord Foundation, 20170843
Note

open access

Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2021-06-29Bibliographically approved
2. A characterization of graded von Neumann regular rings with applications to Leavitt path algebras
Open this publication in new window or tab >>A characterization of graded von Neumann regular rings with applications to Leavitt path algebras
2021 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 567, p. 91-113Article in journal (Refereed) Published
Abstract [en]

We prove a new characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are graded von Neumann regular. More precisely, we show that a Leavitt path algebra LR(E) with coefficients in a unital ring R is graded von Neumann regular if and only if R is von Neumann regular. We also prove that both Leavitt path algebras and corner skew Laurent polynomial rings over von Neumann regular rings are semiprimitive and semiprime. Thereby, we generalize a result by Abrams and Aranda Pino on the semiprimitivity of Leavitt path algebras over fields. © 2020 The Author(s)

Place, publisher, year, edition, pages
Academic Press Inc., 2021
Keywords
Corner skew Laurent polynomial ring, Epsilon-strongly graded ring, Leavitt path algebra, Partial crossed product, Von Neumann regular ring
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-20529 (URN)10.1016/j.jalgebra.2020.09.022 (DOI)000590242700005 ()2-s2.0-85091631468 (Scopus ID)
Funder
The Crafoord Foundation, 20170843
Note

open access

Available from: 2020-10-09 Created: 2020-10-09 Last updated: 2021-04-21Bibliographically approved
3. 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
2020 (English)In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 9, no 9, article id 2050162Article in journal (Refereed) Published
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)10.1142/S0219498820501625 (DOI)000563009600001 ()
Funder
The Crafoord Foundation, 20170843
Note

open access

Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2021-10-08Bibliographically approved
4. The graded structure of algebraic Cuntz-Pimsner rings
Open this publication in new window or tab >>The graded structure of algebraic Cuntz-Pimsner rings
2020 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 224, no 9, article id UNSP 106369Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier B.V., 2020
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)10.1016/j.jpaa.2020.106369 (DOI)000526412900011 ()
Funder
The Crafoord Foundation, 20170843
Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2021-04-21Bibliographically approved
5. Prime group graded rings with applications to partial crossed products and Leavitt path algebras
Open this publication in new window or tab >>Prime group graded rings with applications to partial crossed products and Leavitt path algebras
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this article we generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime s-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over s-unital rings; (ii) characterizations of prime s-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterization of prime Leavitt path algebras, by Abrams, Bell and Rangaswamy.

National Category
Algebra and Logic
Research subject
Mathematics and applications
Identifiers
urn:nbn:se:bth-21408 (URN)
Available from: 2021-05-20 Created: 2021-05-20 Last updated: 2021-05-21Bibliographically approved

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