Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896174 | Journal of Algebra | 2018 | 20 Pages |
Abstract
Let α={αg:Rgâ1âRg}gâmor(G) be a partial action of a groupoid G on a (not necessarily associative) ring R and let S=RâαG 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 Rg={0}, for all but finitely many gâmor(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 Rg={0}, for all but finitely many gâmor(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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Patrik Nystedt, Johan Ãinert, Héctor Pinedo,