The Bass and topological stable ranks of the Bohl algebra are infinite

Abstract

The Bohl algebra $\textrm{B}$ is the ring of linear combinations of functions $t^k e^{\lambda t}$, where $k$ is any nonnegative integer, and $\lambda$ is any complex number, with pointwise operations. We show that the Bass stable rank and the topological stable rank of $\textrm{B}$ (where we use the topology of uniform convergence) are infinite.