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On finitely generated modules over quasi-Euclidean rings

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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 \(\text {E}_n(R)\) 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.

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  1. EPFL ENT CBS BBP/HBP, Campus Biotech, B1 Building, Chemin des mines 9, 1202, Geneva, Switzerland

    Luc Guyot

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  1. Luc Guyot
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Correspondence to Luc Guyot.

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The author thanks the Mathematics Institute of Göttingen and the Max Planck Institute for Mathematics in Bonn for the excellent conditions provided for his stay at these institutions, during which the paper was written.

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Guyot, L. On finitely generated modules over quasi-Euclidean rings. Arch. Math. 108, 357–363 (2017). https://doi.org/10.1007/s00013-016-1009-9

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  • DOI: https://doi.org/10.1007/s00013-016-1009-9

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