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Mathematics of Computation

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Stably free modules over $\mathbf {R}{[X]}$ of rank $> \dim \mathbf {R}$ are free


Author: Ihsen Yengui
Journal: Math. Comp. 80 (2011), 1093-1098
MSC (2010): Primary 13C10, 19A13, 14Q20, 03F65
DOI: https://doi.org/10.1090/S0025-5718-2010-02427-5
Published electronically: September 27, 2010
MathSciNet review: 2772113
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Abstract: We prove that for any finite-dimensional ring $\mathbf {R}$ and $n\geq \dim \mathbf {R} +2$, the group $\textrm {E}_{n}(\textbf {R}[X])$ acts transitively on $\textrm {Um}_{n}(\mathbf {R}[X])$. In particular, we obtain that for any finite-dimensional ring $\mathbf {R}$, all finitely generated stably free modules over $\mathbf {R}[X]$ of rank $> \dim \mathbf {R}$ are free. This result was only known for Noetherian rings. The proof we give is short, simple, and constructive.


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Additional Information

Ihsen Yengui
Affiliation: Department of Mathematics, Faculty of Sciences of Sfax, 3000 Sfax, Tunisia
MR Author ID: 657905
Email: ihsen.yengui@fss.rnu.tn

Keywords: Stably free modules, unimodular vectors, Quillen-Suslin theorem, Hermite rings, Hermite ring conjecture, constructive mathematics
Received by editor(s): June 13, 2009
Published electronically: September 27, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.