On prime non-primitive von Neumann regular algebras

TL;DR

This paper classifies directed graphs based on when their associated Leavitt path algebras are primitive, leading to examples of von Neumann regular prime rings that are not primitive.

Contribution

It provides a classification criterion for graphs whose Leavitt path algebras are primitive, revealing new examples of von Neumann regular prime rings that lack primitivity.

Findings

Identified conditions on graphs for Leavitt path algebras to be primitive

Constructed examples of von Neumann regular prime rings that are not primitive

Extended understanding of the structure of Leavitt path algebras

Abstract

Let $E$ be any directed graph, and $K$ any field. We classify those graphs $E$ for which the Leavitt path algebra $L_K(E)$ is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive.