Simple flat Leavitt path algebras are von Neumann regular

TL;DR

This paper proves that simple flat modules in certain algebraic structures imply the entire ring is von Neumann regular, specifically for Leavitt path algebras, resolving a long-standing open question.

Contribution

It demonstrates that SF Leavitt path algebras are von Neumann regular, providing a positive answer to Ramamurthi's question for this class.

Findings

SF Steinberg algebras have an aperiodic unit space

Graphs associated with these algebras are acyclic

SF Leavitt path algebras are von Neumann regular

Abstract

For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40 years ago who called such rings SF-rings (i.e., simple modules are flat). In this note we show that a SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids this implies that the graphs are acyclic. Combining with the Abrams-Rangaswamy Theorem, it follows that SF Leavitt path algebras are regular, answering Ramamurthi's question in positive for the class of Leavitt path algebras.