Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras

TL;DR

This paper explores the ring-theoretic properties of partial skew groupoid rings derived from groupoid actions and applies these findings to characterize Leavitt path algebras, providing new insights into their structure.

Contribution

It establishes a connection between partial skew groupoid rings and Leavitt path algebras, characterizing when the actions are group-type and deriving properties of the algebras.

Findings

Ring-theoretic properties of $A$ and $B$ are related.

Characterization of group-type partial actions on Leavitt path algebras.

Examples illustrating the theoretical results.

Abstract

Let $\alpha=(A_g,\alpha_g)_{g\in G}$ be a group-type partial action of a connected groupoid $G$ on a ring $A=\bigoplus_{z\in G_0}A_z$ and $B=A\star_{\alpha}G$ the corresponding partial skew groupoid ring. In the first part of this paper we investigate the relation of several ring theoretic properties between $A$ and $B$. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action $\lambda$, we characterize when $\lambda$ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general partial skew groupoid rings. Several examples that illustrate the results on Leavitt path algebras are presented.