# Group-graded rings satisfying the strong rank condition

## TL;DR

This paper characterizes when strongly graded rings satisfy the strong rank condition, linking it to the amenability of the grading group and properties of the base ring, with implications for group rings and von Neumann algebras.

## Contribution

It establishes a new criterion for SRC in strongly graded rings based on base ring SRC and group amenability, and proves a conjecture relating group von Neumann algebras to amenability.

## Key findings

- R satisfies SRC iff R_1 satisfies SRC and G is amenable
- Characterization of amenability via group von Neumann algebra
- Applications to group rings and module theory

## Abstract

A ring $R$ satisfies the {\it strong rank condition} (SRC) if, for every natural number $n$, the free $R$-submodules of $R^n$ all have rank $\leq n$. Let $G$ be a group and $R$ a ring strongly graded by $G$ such that the base ring $R_1$ is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that $R$ satisfies SRC if and only if $R_1$ satisfies SRC and $G$ is amenable. The special case of this result for group rings allows us to prove a characterization of amenability involving the group von Neumann algebra that was conjectured by Wolfgang L\"uck. In addition, we include two applications to the study of group rings and their modules.