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

Abstract

We prove a new 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 LR(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.