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Publications (4 of 4) Show all publications
Lännström, D. (2019). Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras. Algebras and Representation Theory
Open this publication in new window or tab >>Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras
2019 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079Article in journal (Refereed) Epub ahead of print
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, 2019
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-18607 (URN)10.1007/s10468-019-09909-0 (DOI)2-s2.0-85068819896 (Scopus ID)
Available from: 2019-09-10 Created: 2019-09-10 Last updated: 2019-09-10
Lännström, D.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
Lännström, D.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
Lännström, D.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
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-8445-3936

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