Leavitt path algebras which are Zorn rings

TL;DR

This paper characterizes when Leavitt path algebras over a field are Zorn rings, linking algebraic properties to graph conditions, and explores related properties of graph C*-algebras.

Contribution

It provides a complete characterization of Zorn rings among Leavitt path algebras based on graph Condition (L), and examines the weak regularity and C*-algebra analogs.

Findings

Leavitt path algebra is a Zorn ring iff the graph satisfies Condition (L).

Leavitt path algebra is weakly regular iff all its homomorphic images are Zorn rings.

The paper extends the analysis to graph C*-algebras.

Abstract

Let E be an arbitrary directed graph and let K be any field. It is shown that the Leavitt path algebra A of the graph E over the field K is a Zorn ring if and only if the graph E satisfies the Condition (L), that is, every cycle in E has an exit. It is also shown that the Leavitt path algebra A is a weakly regular ring if and only if every homomorphic image of A is a Zorn ring. The corresponding statement for graph C*-algebras is also investigated.