Finitely generated modules over quasi-Euclidean rings

TL;DR

This paper investigates the action of elementary matrix groups on unimodular rows over quasi-Euclidean rings, establishing transitivity conditions and classifying generating tuples in finitely generated modules.

Contribution

It extends the classification of generating tuples to modules over quasi-Euclidean rings, showing transitivity of elementary group actions and linking unimodular row orbits to the ring's unit group.

Findings

Action of $E_n(R)$ is transitive on unimodular rows when $n > k$

Classification of generating tuples in finitely generated modules over quasi-Euclidean rings

Connection between unimodular row orbits and the unit group of quotient rings

Abstract

Let R be a unital commutative ring and let $M$ be an $R$-module that is generated by $k$ elements but not less. Let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by the elementary matrices. In this paper we study the action of $E_n(R)$ by matrix multiplication on the set $Um_n(M)$ of unimodular rows of $M$ of length $n \ge k$. Assuming $R$ is moreover Noetherian and quasi-Euclidean, e.g., $R$ is a direct sum of finitely many Euclidean rings, we show that this action is transitive if $n > k$. We also prove that $Um_k(M) /E_k(R)$ is equipotent with the unit group of $R/(a_1)$ where $(a_1)$ 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.