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Artinian and noetherian partial skew groupoid rings
University West, SWE.ORCID iD: 0000-0001-6594-7041
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.ORCID iD: 0000-0001-8095-0820
Industrial University of Santander, COL.
2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 503, p. 433-452Article in journal (Refereed) Published
Abstract [en]

Let α={α_g : R_{g^{−1}}→R_g}_{g∈mor(G)} be a partial action of a groupoid G on a (not necessarily associative) ring R and let S=R⋆G be the associated partial skew groupoid ring. We show that if α is global and unital, then S is left (right) artinian if and only if R is left (right) artinian and R_g={0}, for all but finitely many g∈mor(G). We use this result to prove that if α is unital and R is alternative, then S is left (right) artinian if and only if R is left (right) artinian and R_g={0}, for all but finitely many g∈mor(G). This result applies to partial skew group rings, in particular. Both of the above results generalize a theorem by J. K. Park for classical skew group rings, i.e. the case when R is unital and associative, and G is a group which acts globally on R. We provide two additional applications of our main results. Firstly, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (not necessarily associative) groupoid rings. This result is in turn applied to partial group algebras. Secondly, we give a characterization of artinian Leavitt path algebras. At the end of the article, we relate noetherian and artinian properties of partial skew groupoid rings to those of global skew groupoid rings, as well as establish two Maschke-type results, thereby generalizing results by M. Ferrero and J. Lazzarin for partial skew group rings to the case of partial skew groupoid rings.

Place, publisher, year, edition, pages
Academic Press, 2018. Vol. 503, p. 433-452
Keywords [en]
artinian ring, partial skew groupoid ring, partial skew group ring, partial group algebra, Leavitt path algebra
National Category
Algebra and Logic Other Mathematics
Identifiers
URN: urn:nbn:se:bth-11699DOI: 10.1016/j.jalgebra.2018.02.007ISI: 000429764400020OAI: oai:DiVA.org:bth-11699DiVA, id: diva2:910158
Available from: 2016-03-08 Created: 2016-03-08 Last updated: 2018-04-26Bibliographically approved

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Öinert, Johan

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