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Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.ORCID iD: 0000-0001-8445-3936
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 [en]
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: urn:nbn:se:bth-18607DOI: 10.1007/s10468-019-09909-0Scopus ID: 2-s2.0-85068819896OAI: oai:DiVA.org:bth-18607DiVA, id: diva2:1349795
Available from: 2019-09-10 Created: 2019-09-10 Last updated: 2019-09-10

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