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Artinian and noetherian partial skew groupoid rings
University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.ORCID iD: 0000-0001-6594-7041
Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, Sweden.
Universidad Industrial de Santander, Escuela de Matemáticas, Carrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678, Bucaramanga, Colombia.
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 alpha = {alpha(g) : Rg-1 -> R-g}(g is an element of mor(G)) be a partial action of a groupoid G on a (not necessarily associative) ring R and let S = R-star alpha G be the associated partial skew groupoid ring. We show that if a 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 is an element of mor(G). We use this result to prove that if a 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 is an element of 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 Inc. , 2018. Vol. 503, p. 433-452
National Category
Mathematics
Research subject
ENGINEERING, Mathematics
Identifiers
URN: urn:nbn:se:hv:diva-12244DOI: 10.1016/j.jalgebra.2018.02.007ISI: 000429764400020Scopus ID: 2-s2.0-85044284762OAI: oai:DiVA.org:hv-12244DiVA, id: diva2:1196328
Note

Available online 14 February 2018

Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-26Bibliographically approved

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