# Group ring with infinite stable rank

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?

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.