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Artinian partial skew groupoid rings
University West, Sweden.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, Colombia.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let α={α_g : R_{g^{−1}}→R_g}_{g∈mor(G)} be a partial action of a groupoid G on a non-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 Ris left (right) artinian and R_g={0}, for all but finitely many g∈mor(G). Both of these results apply to partial skew group rings, and in particular they generalize a result 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. Moreover, we provide two applications of our main result. First, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (non-associative) groupoid rings. This result is in turn applied to partial group algebras. Finally, we give a characterization of artinian Leavitt path algebras.

Keyword [en]
artinian ring, partial skew groupoid ring, partial skew group ring, partial group algebra, Leavitt path algebra
National Category
Algebra and Logic Other Mathematics
URN: urn:nbn:se:bth-11699OAI: diva2:910158
Available from: 2016-03-08 Created: 2016-03-08 Last updated: 2016-03-10Bibliographically approved

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Nystedt, PatrikÖinert, Johan
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