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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
##### 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
Open this publication in new window or tab >>Induced quotient group gradings of epsilon-strongly graded rings
##### 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, Ã–inert 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. Journal of Pure and Applied Algebra
Open this publication in new window or tab >>The graded structure of algebraic Cuntz-Pimsner rings
(English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376Article in journal (Refereed) Epub ahead of print
##### 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.

Elsevier B.V.
##### 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)
##### Funder
The Crafoord Foundation, 20170843 Available from: 2019-04-11 Created: 2019-04-11 Last updated: 2020-03-20Bibliographically approved
##### Identifiers
ORCID iD: orcid.org/0000-0001-8445-3936

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