Stable finiteness of endomorphism rings

TL;DR

This paper proves that for a left Noetherian ring and a sofic group, the group ring is stably finite, using a novel combination of combinatorial and localization techniques.

Contribution

It provides a new, self-contained proof of the stable finiteness of group rings over Noetherian rings for sofic groups, advancing understanding in algebraic structures.

Findings

Group rings over Noetherian rings are stably finite for sofic groups.

The proof combines combinatorial ideas with localization theory.

The approach simplifies existing proofs and is self-contained.

Abstract

We combine a combinatorial idea of Benjamin Weiss and some localization theory of Grothendieck categories to give a short and completely self-contained proof of the following recent result of Hanfeng Li and Bingbing Liang: Given a left Noetherian ring $R$ and a sofic group $G$, the group-ring $R[G]$ is stably finite.