Leavitt path algebras are graded von Neumann regular rings

TL;DR

This paper proves that all Leavitt path algebras are graded von Neumann regular rings, extending previous results limited to acyclic graphs, and explores their algebraic properties.

Contribution

It establishes that Leavitt path algebras of any graph are graded von Neumann regular, broadening the class beyond acyclic graphs and deriving related algebraic properties.

Findings

Leavitt path algebras are graded von Neumann regular for all graphs.

Several algebraic properties follow from this regularity, such as radical triviality and module flatness.

Finitely generated graded ideals are generated by idempotents.

Abstract

In sharp contrast to the Abrams-Rangaswamy Theorem that the only von Neumann regular Leavitt path algebras are exactly those associated to acyclic graphs, here we prove that the Leavitt path algebra of any arbitrary graph is a graded von Neumann regular ring. Several properties of Leavitt path algebras, such as triviality of the Jacobson radical, flatness of graded modules and finitely generated graded right (left) ideals being generated by an idempotent element, follow as a consequence of general theory of grade von Neumann regular rings.