A characterization of graded von Neumann regular rings with applications to Leavitt path algebras

TL;DR

This paper characterizes graded von Neumann regular rings using nearly epsilon-strongly graded rings and applies this to show Leavitt path algebras over rings are graded von Neumann regular if and only if the coefficient ring is von Neumann regular.

Contribution

It provides a new characterization of graded von Neumann regular rings and generalizes existing results on Leavitt path algebras over fields to those over rings.

Findings

Leavitt path algebra $L_R(E)$ is graded von Neumann regular iff $R$ is von Neumann regular.

Leavitt path algebras over von Neumann regular rings are semiprimitive and semiprime.

Generalizes semiprimitivity results of Leavitt path algebras over fields.

Abstract

We provide a characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are graded von Neumann regular. More precisely, we show that a Leavitt path algebra $L_R(E)$ with coefficients in a unital ring $R$ is graded von Neumann regular if and only if $R$ is von Neumann regular. We also prove that both Leavitt path algebras and corner skew Laurent polynomial rings over von Neumann regular rings are semiprimitive and semiprime. Thereby, we generalize a result by Abrams and Aranda Pino on the semiprimitivity of Leavitt path algebras over fields.