On finitely generated modules over quasi-Euclidean rings

Let R be a unital commutative ring, and let M be an R-module that is generated by k elements but not less. Let be the subgroup of generated by the elementary matrices. In this paper we study the action of by matrix multiplication on the set of unimodular rows of M of length . Assuming R is moreover Noetherian and quasi-Euclidean, e.g., R is a direct product of finitely many Euclidean rings, we show that this action is transitive if . We also prove that is equipotent with the unit group of where is the first invariant factor of M. These results encompass the well-known classification of Nielsen non-equivalent generating tuples in finitely generated Abelian groups.