The Bass and topological stable ranks of and

Abstract

In this note we prove that the Bass stable rank of is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of is two and the topological stable rank of is two, settling a conjecture by the second author. We also present a -free proof of the second author's characterization of the reducible pairs in .