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arXiv · 1604.07813

Finitely generated modules over quasi-Euclidean rings

Luc Guyot
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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.

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Luc Guyot. 2016-04-26. Finitely generated modules over quasi-Euclidean rings. https://doi.org/10.1007/s00013-016-1009-9

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