Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

  • Daniel LännströmEmail author
Open Access


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 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.


Group graded ring Epsilon-strongly graded ring Chain conditions Leavitt path algebra Partial crossed product 

Mathematics Subject Classification (2010)

16P20 16P40 16W50 16S35 



This research was partially supported by the Crafoord Foundation (grant no. 20170843). The author is grateful to Johan Öinert, Stefan Wagner and Patrik Nystedt for giving comments and feedback that helped to improve this manuscript.


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Authors and Affiliations

  1. 1.Department of Mathematics and Natural SciencesBlekinge Institute of TechnologyKarlskronaSweden

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