Artinian and noetherian partial skew groupoid rings

JOURNAL OF ALGEBRA

卷:503

Article

Nystedt, Patrik¹  Oinert, Johan²  Pinedo, Hector³ 

[1] Univ West, Dept Engn Sci, SE-46186 Trollhattan, Sweden

[2] Blekinge Inst Technol, Dept Math & Nat Sci, SE-37179 Karlskrona, Sweden

[3] Univ Ind Santander, Escuela Matemat, Carrera 27 Calle 9, Bucaramanga, Colombia

关键词: Artinian ring;    Noetherian ring;    Partial skew groupoid ring;    Partial skew group ring;    Partial group algebra;    Leavitt path algebra;    Globalization;    Morita equivalence;   

DOI  :  10.1016/j.jalgebra.2018.02.007

来源: Elsevier

【 摘 要 】

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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free