The realization problem for von Neumann regular rings

TL;DR

This paper surveys recent advances in the realization problem for von Neumann regular rings, focusing on whether certain monoids can be represented as classes of finitely generated projective modules over such rings.

Contribution

It provides a comprehensive overview of recent progress and results related to the realization problem in the context of von Neumann regular rings.

Findings

Progress on the realization problem for countable conical refinement monoids

Conditions under which monoids can be realized as projective module classes

Open questions and future directions in the field

Abstract

We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right $R$-modules over a von Neumann regular ring $R$.