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2 results on '"Ihsen Yengui"'

1. The syzygy theorem for Bézout rings

Author

Stefan Neuwirth, Ihsen Yengui, Henri Lombardi, Maroua Gamanda, Département de Mathematiques [Sfax], Faculté des Sciences de Sfax, Université de Sfax - University of Sfax-Université de Sfax - University of Sfax, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), John Templeton Foundation (ID 60842), ANR-15-IDEX-0003,BFC,ISITE ' BFC(2015), Fédération Bourgogne Franche-Comté Mathématiques (BFC-Math ), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), and ANR: 15-IDEX-0003,BFC,ISITE « BFC(2015)

Subjects
Pure mathematics, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], Schreyer's syzygy algorithm, 010103 numerical & computational mathematics, 01 natural sciences, Constructive, Valuation ring, MSC 2010: 13D02, 13P10, 13C10, 13P20, 14Q20, Finitely-generated abelian group, 0101 mathematics, Syzygy theorem, Gröbner ring, Monomial order, strict Bézout ring, Mathematics, Algebra and Number Theory, Hilbert's syzygy theorem, Mathematics::Commutative Algebra, monomial order, valuation ring, Applied Mathematics, 010102 general mathematics, free resolution, Divisibility rule, Mathematics - Commutative Algebra, 16. Peace & justice, dynamical Gröbner basis, 13D02, 13P10, 13C10, 13P20, 14Q20, Schreyer's monomial order, Computational Mathematics
Abstract

International audience; We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

Published
2019
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2. Stably free modules over $\mathbf{R}[X]$ of rank $> \dim\mathbf{R}$ are free

Author

Ihsen Yengui

Subjects
Noetherian, Discrete mathematics, Ring (mathematics), Algebra and Number Theory, Mathematics::Commutative Algebra, Group (mathematics), Applied Mathematics, Combinatorics, Quillen–Suslin theorem, Computational Mathematics, Simple (abstract algebra), Rank (graph theory), Finitely-generated abelian group, Mathematics
Abstract

We prove that for any finite-dimensional ring R and n > dim R+2, the group E n (R[X]) acts transitively on Um n (R[X]). In particular, we obtain that for any finite-dimensional ring R, all finitely generated stably free modules over R[X] of rank > dim R are free. This result was only known for Noetherian rings. The proof we give is short, simple, and constructive.

Published
2011
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