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

Abstract

Let $\alpha =(A_g,{\alpha }_g{)}_{g\in G}$ be a group-type partial action of a connected groupoid $G$ on a ring $A={⨁}_{z\in G_0}{A}_z$ 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.