In searching for a counterexample in homological stability, I came across the following question:



> 
> Is there a known example of a finitely presented group $G$, so that the group ring $\mathbb{Z}[G]$ has infinite Bass [stable rank](https://encyclopediaofmath.org/wiki/Stable_rank)?
> 
> 
> 



Group ring with infinite stable rank
**Yes**, the integral group ring $\mathbb{Z}[F\_2]$ of the free group $F\_2$ on two generators has **infinite** stable rank.


This can be deduced from [1, Corollary 3.6]:



> 
> There exists a cyclic $\mathbb{Z}[F\_2]$-module $M$ with the following property.
> For every $N \ge 1$, there exists an epimorphism $\theta\_N: (\mathbb{Z}[F\_2])^N \twoheadrightarrow M$ such that $(\mathbb{Z}[F\_2])^N$ cannot be generated by $N$ elements one of which is contained in $\ker \theta\_N$.
> 
> 
> 


It should be easy to see that the cyclic $\mathbb{Z}[F\_2]$-module $M$ above has infinite stable rank. (Actually, explicit non-stable unimodular rows of length $N$ for every $N \ge 2$ can be extracted from the proof).


Then combine the previous result with [2, Lemma 11.4.6]



> 
> If $n$ is in the stable range of a finitely generated right $R$-module $M$ then $n$ is in the stable range of any factor module $M/K$.
> 
> 
> 


to conclude that the stable rank of $\mathbb{Z}[F\_2]$ is infinite.




---


[1] M. Evans, "Presentations of groups involving more generators than necessary", 1992.  

[2] J. McConnell and J. Robson, "Noncommutative Noethering rings", 1987.


