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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Stably free modules over $\mathbf {R}{[X]}$ of rank $> \dim \mathbf {R}$ are free
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by Ihsen Yengui PDF
Math. Comp. 80 (2011), 1093-1098 Request permission

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
  • Received by editor(s): June 13, 2009
  • Published electronically: September 27, 2010
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2772113