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

Abstract

Let α = (Ag, αg)g∈G be a group-type partial action of a connected groupoid G on a ring A = z∈G 0 Az and B := A ⋆α 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 λ, we characterize when λ 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.

On groupoid graded von Neumann regular rings and a Brandt groupoid graded Leavitt path algebras

“…Namely, in [18], the Leavitt path algebra L K (E) is realized as a partial skew groupoid ring D(X) ⋆ λ G(E), where D(X) is a certain K-algebra and λ is a partial action of the free path groupoid G(E) on D(X). Under some restrictions, in [12], the G(E)-graded von Neumann regularity of L K (E) is characterized in terms of D(X). Namely, let E be a connected graph and let the set of vertices E 0 be finite.…”

“…As a corollary, one also obtains that an epsilon-strongly graded ring R = g∈G R g is graded von Neumann regular if and only if R e is von Neumann regular (Corollary 3.11 in [46]). For results concerning rings graded by finite small categories all of whose morphisms are invertible, the reader is referred to [12,52].…”