Artinian and noetherian partial skew groupoid rings

TL;DR

This paper characterizes when partial skew groupoid rings are artinian, extending classical results to non-associative rings, partial actions, and groupoids, with applications to Leavitt path algebras and groupoid rings.

Contribution

It provides new criteria for artinianity of partial skew groupoid rings, generalizing known results to non-associative rings and groupoids, and explores their applications.

Findings

Characterization of artinian partial skew groupoid rings.

Extension of classical results to non-associative and partial actions.

Applications to Leavitt path algebras and groupoid rings.

Abstract

Let $\alpha = \{ \alpha_g : R_{g^{-1}} \rightarrow R_g \}_{g \in \textrm{mor}(G)}$ be a partial action of a groupoid $G$ on a non-associative ring $R$ and let $S = R \star_{\alpha} G$ be the associated partial skew groupoid ring. We show that if $\alpha$ 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 $g \in \textrm{mor}(G)$. We use this result to prove that if $\alpha$ 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 $g \in \textrm{mor}(G)$. 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. Firstly, 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. Secondly, we give a characterization of artinian Leavitt path algebras. At the end of the article, we use globalization to analyse noetherianity and artinianity of partial skew groupoid rings as well as establishing two Maschke-type results, thereby generalizing results by Ferrero and Lazzarin from the group graded case to the groupoid situation.