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 \(\text {E}_n(R)\) be the subgroup of \(\text {GL}_n(R)\) generated by the elementary matrices. In this paper we study the action of \(\text {E}_n(R)\) by matrix multiplication on the set \(\text {Um}_n(M)\) of unimodular rows of M of length \(n \ge k\). 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 \(n > k\). We also prove that \(\text {Um}_k(M) /\text {E}_k(R)\) is equipotent with the unit group of \(R/\mathfrak {a}_1\) where \(\mathfrak {a}_1\) 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.