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Projective Modules Over Polynomial Rings

  • Ihsen Yengui
Part of the Lecture Notes in Mathematics book series (LNM, volume 2138)

Keywords

Prime Ideal Polynomial Ring Projective Module Free Module Valuation Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. [1]
    Abedelfatah, A.: On stably free modules over Laurent polynomial rings. Proc. Am. Math. Soc. 139, 4199–4206 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Abedelfatah, A.: On the action of the elementary group on the unimodular rows. J. Algebra 368, 300–304 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Adams, W.W., Loustaunau, P.: An Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)Google Scholar
  4. [4]
    Amidou, M., Yengui, I.: An algorithm for unimodular completion over Laurent polynomial rings. Linear Algebra Appl. 429, 1687–1698 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Arnold, E.A.: Modular algorithms for computing Gröbner bases. J. Symb. Comput. 35, 403–419 (2003)zbMATHCrossRefGoogle Scholar
  6. [6]
    Aschenbrenner, M.: Ideal membership in polynomial rings over the integers. J. Am. Math. Soc. 17, 407–441 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Ayoub, C.: On constructing bases for ideals in polynomial rings over the integers. J. Number Theory 17(2), 204–225 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Barhoumi, S.: Seminormality and polynomial ring. J. Algebra 322, 1974–1978 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Barhoumi, S., Lombardi, H.: An algorithm for the Traverso-Swan theorem over seminormal rings. J. Algebra 320, 1531–1542 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Barhoumi, S., Yengui, I.: On a localization of the Laurent polynomial ring. JP. J. Algebra Number Theory Appl. 5(3), 591–602 (2005)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Barhoumi, S., Lombardi, H., Yengui, I.: Projective modules over polynomial rings: a constructive approach. Math. Nach 282, 792–799 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Bass, H.: Algebraic K-Theory. W.A. Benjamin Inc., New York/Amsterdam (1968)zbMATHGoogle Scholar
  13. [13]
    Bass, H.: Libération des modules projectifs sur certains anneaux de polynômes, Sém. Bourbaki 1973/74, exp. 448. Lecture Notes in Mathematics, vol. 431, pp. 228–254. Springer, Berlin/ New York (1975)Google Scholar
  14. [14]
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, 2nd edn. Springer, Berlin (2006)Google Scholar
  15. [15]
    Bayer, D.: The division algorithm and the Hilbert scheme. Ph.D. dissertation, Harvard University (1982)Google Scholar
  16. [16]
    Bernstein, D.: Fast ideal arithmetic via lazy localization. In: Cohen, H. (ed.) Algorithmic Number Theory. Proceeding of the Second International Symposium, ANTS-II, Talence, France, 18–23 May 1996. Lecture Notes in Computer Science, vol. 1122, pp. 27–34. Springer, Berlin (1996)Google Scholar
  17. [17]
    Bernstein, D.: Factoring into coprimes in essentially linear time. J. Algorithms 54, 1–30 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Boileau, A., Joyal, A.: La Logique des Topos. J. Symb. Log. 46, 6–16 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Bourbaki, N.: Algèbre Commutative. Chapitres 5–6. Masson, Paris (1985)Google Scholar
  20. [20]
    Brewer, J., Costa, D.: Projective modules over some non-Noetherian polynomial rings. J. Pure Appl. Algebra 13, 157–163 (1978)MathSciNetCrossRefGoogle Scholar
  21. [21]
    Brickenstein, M., Dreyer, A., Greuel, G.-M., Wedler, M., Wienand, O.: New developments in the theory of Groebner bases and applications to formal verification. J. Pure Appl. Algebra 213, 1612–1635 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen polynomideal. Ph.D. thesis, University of Innsbruck (1965)Google Scholar
  23. [23]
    Buchberger, B.: A critical pair/completion algorithm for finitely-generated ideals in rings. In: Logic and Machines: Decision Problems and Complexity. Springer Lectures Notes in Computer Science, vol. 171, pp. 137–161. Springer, New York (1984)Google Scholar
  24. [24]
    Buchmann, J., Lenstra, H.: Approximating rings of integers in number fields. J. Théor. Nombres Bordeaux 6(2), 221–260 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Byrne, E., Fitzpatrick, P.: Gröbner bases over Galois rings with an application to decoding alternant codes. J. Symb. Comput. 31, 565–584 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Cahen, P.-J.: Construction B,I,D et anneaux localement ou résiduellement de Jaffard. Archiv. Math. 54, 125–141 (1990)Google Scholar
  27. [27]
    Cahen, P.-J., Elkhayyari, Z., Kabbaj, S.: Krull and valuative dimension of the Serre conjecture ring \(R\langle n\rangle\). In: Commutative Ring Theory. Lecture Notes in Pure and Applied Mathematics, vol. 185, pp. 173–180. Marcel Dekker, New York (1997)Google Scholar
  28. [28]
    Cai, Y., Kapur, D.: An algorithm for computing a Gröbner basis of a polynomial ideal over a ring with zero divisors. Math. Comput. Sci. 2, 601–634 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    Caniglia, L., Cortiñas, G., Danón, S., Heintz, J., Krick, T., Solernó, P.: Algorithmic aspects of Suslin’s proof of Serre’s conjecture. Comput. Complex. 3, 31–55 (1993)zbMATHCrossRefGoogle Scholar
  30. [30]
    Cohn, P.M.: On the structure of GLn of a ring. Publ. Math. I.H.E.S. 30, 5–54 (1966)Google Scholar
  31. [31]
    Coquand, T.: Sur un théorème de Kronecker concernant les variétés algébriques. C. R. Acad. Sci. Paris, Ser. I 338, 291–294 (2004)MathSciNetzbMATHGoogle Scholar
  32. [32]
    Coquand, T.: On seminormality. J. Algebra 305, 577–584 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    Coquand, T.: A refinement of Forster’s theorem. Preprint (2007)Google Scholar
  34. [34]
    Coquand, T., Lombardi, H.: Hidden constructions in abstract algebra (3) Krull dimension of distributive lattices and commutative rings. In: Fontana, M., Kabbaj, S.-E., Wiegand, S. (eds.) Commutative Ring Theory and Applications. Lecture Notes in Pure and Applied Mathematics, vol. 131, pp. 477–499. Marcel Dekker, New York (2002)Google Scholar
  35. [35]
    Coquand, T., Lombardi, H.: A short proof for the Krull dimension of a polynomial ring. Am. Math. Mon. 112, 826–829 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Coquand, T., Quitté, C.: Constructive finite free resolutions. Manuscripta Math. 137, 331–345 (2011)CrossRefGoogle Scholar
  37. [37]
    Coquand, T., Ducos, L., Lombardi, H., Quitté, C.: L’idéal des coefficients du produit de deux polynômes. Rev. Math. Enseign. Supér. 113, 25–39 (2003)Google Scholar
  38. [38]
    Coquand, T., Lombardi, H., Quitté, C.: Generating nonnoetherian modules constructively. Manuscripta Math. 115, 513–520 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    Coquand, T., Lombardi, H., Roy, M.-F.: An elementary characterisation of Krull dimension. In: Corsilla, L., Schuster, P. (eds.) From Sets and Types to Analysis and Topology: Towards Practicable Foundations for Constructive Mathematics. Oxford University Press, Oxford (2005)Google Scholar
  40. [40]
    Coquand, T., Lombardi, H., Schuster, P.: A nilregular element property. Arch. Math. 85, 49–54 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    Coquand, T., Lombardi, H., Schuster, P.: The projective spectrum as a distributive lattice. Cahiers de Topologie et Géométrie différentielle catégoriques 48, 220–228 (2007)MathSciNetzbMATHGoogle Scholar
  42. [42]
    Coste, M., Lombardi, H., Roy, M.-F.: Dynamical method in algebra: effective Nullstellensätze. Ann. Pure Appl. Logic 111, 203–256 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms, 2nd edn. Springer, New York (1997)Google Scholar
  44. [44]
    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: SINGULAR 4-0-2 – A Computer Algebra System for Polynomial Computations. http://www.singular.uni-kl.de (2015)
  45. [45]
    Decker, W., Pfister, G.: A First Course in Computational Algebraic Geometry. AIMS Library Series. Cambridge University Press, Cambridge (2013)zbMATHCrossRefGoogle Scholar
  46. [46]
    Della Dora, J., Dicrescenzo, C., Duval, D.: About a new method for computing in algebraic number fields. In: Caviness, B.F. (ed.) EUROCAL ’85. Lecture Notes in Computer Science, vol. 204, pp. 289–290. Springer, Berlin (1985)Google Scholar
  47. [47]
    Diaz-Toca, G.M., Lombardi, H.: Dynamic Galois theory. J. Symb. Comput. 45, 1316–1329 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    Ducos, L., Monceur, S., Yengui, I.: Computing the V-saturation of finitely-generated submodules of V[X]m where V is a valuation domain. J. Symb. Comput. 72, 196–205 (2016)MathSciNetCrossRefGoogle Scholar
  49. [49]
    Ducos, L., Quitté, C., Lombardi, H., Salou, M.: Théorie algorithmique des anneaux arithmétiques, de Prüfer et de Dedekind. J. Algebra 281, 604–650 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    Ducos, L., Valibouze, A., Yengui, I.: Computing syzygies over V[X 1, , X k], V a valuation domain. J. Algebra 425, 133–145 (2015)Google Scholar
  51. [51]
    Duval, D., Reynaud, J.-C.: Sketches and computation (part II) dynamic evaluation and applications. Math. Struct. Comput. Sci. 4, 239–271 (1994) (see http://www.Imc.imag.fr/Imc-cf/Dominique.Duval/evdyn.html)
  52. [52]
    Edwards, H.: Divisor Theory. Birkhäuser, Boston (1989)Google Scholar
  53. [53]
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, New York (1995)zbMATHGoogle Scholar
  54. [54]
    Ebert, G.L.: Some comments on the modular approach to Gröbner-bases. ACM SIGSAM Bull. 17, 28–32 (1983)zbMATHCrossRefGoogle Scholar
  55. [55]
    Ellouz, A., Lombardi, H., Yengui, I.: A constructive comparison of the rings R(X) and \(\mathbf{R}\langle X\rangle\) and application to the Lequain-Simis induction theorem. J. Algebra 320, 521–533 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    Español, L.: Dimensión en álgebra constructiva. Doctoral thesis, Universidad de Zaragoza, Zaragoza (1978)Google Scholar
  57. [57]
    Español, L.: Constructive Krull dimension of lattices. Rev. Acad. Cienc. Zaragoza 37, 5–9 (1982)MathSciNetzbMATHGoogle Scholar
  58. [58]
    Español, L.: Le spectre d’un anneau dans l’algèbre constructive et applications à la dimension. Cahiers de topologie et géométrie différentielle catégorique 24, 133–144 (1983)zbMATHGoogle Scholar
  59. [59]
    Español, L.: Dimension of Boolean valued lattices and rings. J. Pure Appl. Algebra 42, 223–236 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  60. [60]
    Español, L.: Finite chain calculus in distributive lattices and elementary Krull dimension. In: Lamban, L., Romero, A., Rubio, J. (eds.) Contribuciones cientificas en honor de Mirian Andres Gomez Servicio de Publicaciones. Universidad de La Rioja, Logroño (2010)Google Scholar
  61. [61]
    Fabiańska, A.: A Maple QuillenSuslin package. http://wwwb.math.rwth-aachen.de/QuillenSuslin/ (2007)
  62. [62]
    Fabiańska, A., Quadrat, A.: Applications of the Quillen-Suslin theorem to the multidimensional systems theory. INRIA Report 6126 (2007), Published in Gröbner Bases in Control Theory and Signal Processing. In: Park, H., Regensburger, G. (eds.) Radon Series on Computation and Applied Mathematics, vol. 3, pp. 23–106. de Gruyter, Berlin (2007)Google Scholar
  63. [63]
    Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5). In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC (2002)CrossRefGoogle Scholar
  64. [64]
    Fitchas, N., Galligo, A.: Nullstellensatz effectif et conjecture de Serre (Théorème de Quillen-Suslin) pour le calcul formel. Math. Nachr. 149, 231–253 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  65. [65]
    Gallo, S., Mishra, B.: A solution to Kronecker’s problem. Appl. Algebra Eng. Commun. Comput. 5, 343–370 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  66. [66]
    Gilmer, R.: Multiplicative Ideal Theory. Queens Paper in Pure and Applied Mathematics, vol. 90. Marcel Dekker, New York (1992)Google Scholar
  67. [67]
    Glaz, S.: On the weak dimension of coherent group rings. Commun. Algebra 15, 1841–1858 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    Glaz, S.: Commutative Coherent Rings. Lectures Notes in Mathematics, vol. 1371, 2nd edn. Springer, Berlin/Heidelberg/New York (1990)Google Scholar
  69. [69]
    Glaz, S.: Finite conductor properties of R(X) and \(\,\mathbf{R}\langle X\rangle\). In: Ideal Theoretic Methods in Commutative Algebra (Columbia, MO, 1999). Lecture Notes in Pure and Applied Mathematics, vol. 220, pp. 231–249. Marcel Dekker, New York (2001)Google Scholar
  70. [70]
    Glaz, S., Vasconcelos, W.V.: Flat ideals III. Commun. Algebra 12, 199–227 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  71. [71]
    Gräbe, H.: On lucky primes. J. Symb. Comput. 15, 199–209 (1994)CrossRefGoogle Scholar
  72. [72]
    Grayson, D.R., Stillman, M.E.: Macaulay2, A Software System for Research in Algebraic Geometry. Available at http://www.math.uiuc.edu/Macaulay2/
  73. [73]
    Greuel, G.M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer, Berlin/Heidelberg/New York (2002)zbMATHCrossRefGoogle Scholar
  74. [74]
    Greuel, G.-M., Seelisch, F., Wienand, O.: The Gröbner basis of the ideal of vanishing polynomials. J. Symb. Comput. 46, 561–570 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  75. [75]
    Gruson, L., Raynaud, M.: Critères de platitude et de projectivité. Techniques de “platification” d’un module. Invent. Math. 13, 1–89 (1971)Google Scholar
  76. [76]
    Gupta, S.K., Murthy, M.P.: Suslin’S Work on Linear Groups over Polynomial Rings and Serre Problem. Indian Statistical Institute Lecture Notes Series, vol. 8. Macmillan, New Delhi (1980)Google Scholar
  77. [77]
    Hadj Kacem, A., Yengui, I.: Dynamical Gröbner bases over Dedekind rings. J. Algebra 324, 12–24 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  78. [78]
    Havas, G., Majewski B.S.: Extended gcd calculation. In: Proceedings of the Twenty-sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995). Congr. Numer. 111, 104–114 (1995)Google Scholar
  79. [79]
    Heinzer, W., Papick, I.J.: Remarks on a remark of Kaplansky. Proc. Am. Math. Soc. 105, 1–9 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  80. [80]
    Huckaba, J.: Commutative Rings with Zero-Divisors. Marcel Dekker, New York (1988)zbMATHGoogle Scholar
  81. [81]
    Hurwitz, A.: Ueber einen Fundamentalsatz der arithmetischen Theorie der algebraischen Größen, pp. 230–240. Nachr. kön Ges. Wiss., Göttingen (1895) [Werke, vol. 2, pp. 198–207]Google Scholar
  82. [82]
    Jambor, S.: Computing minimal associated primes in polynomial rings over the integers. J. Symb. Comput. 46, 1098–1104 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  83. [83]
    Joyal, A.: Spectral spaces and distibutive lattices. Not. Am. Math. Soc. 18, 393 (1971)Google Scholar
  84. [84]
    Joyal, A.: Le théorème de Chevalley-Tarski. Cahiers de Topologie et Géomérie Différentielle 16, 256–258 (1975)Google Scholar
  85. [85]
    Kapur, D., Narendran, P.: An equational approach to theoretical proving in first-order predicate calculus. In: Proceedings of the International Joint Conference on Artificial Intelligence, IJCAI, pp. 1146–1153 (1985)Google Scholar
  86. [86]
    Kandry-Rody, A., Kapur, D.: Computing a Gröbner basis of a polynomial ideal over a euclidean domain. J. Symb. Comput. 6, 37–57 (1988)CrossRefGoogle Scholar
  87. [87]
    Kemper, G.: A Course in Commutative Algebra. Graduate Texts in Mathematics. Springer, Berlin (2011)zbMATHCrossRefGoogle Scholar
  88. [88]
    Kreuzer, M., Robbianno, L.: Computational Commutative Algebra, vol. 2. Springer, Berlin (2005)zbMATHGoogle Scholar
  89. [89]
    Kronecker, L.: Zur Theorie der Formen höherer Stufen Ber, pp. 957–960. K. Akad. Wiss. Berlin (1883) [Werke 2, 417–424]Google Scholar
  90. [90]
    Kunz, E.: Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser, Basel (1991)Google Scholar
  91. [91]
    Lam, T.Y.: Serre’s Conjecture. Lecture Notes in Mathematics, vol. 635. Springer, Berlin/ New York (1978)Google Scholar
  92. [92]
    Lam, T.Y.: Serre’s Problem on Projective Modules. Springer Monographs in Mathematics. Springer, Berlin (2006)CrossRefGoogle Scholar
  93. [93]
    Laubenbacher, R.C., Woodburn, C.J.: An algorithm for the Quillen-Suslin theorem for monoid rings. J. Pure Appl. Algebra 117/118, 395–429 (1997)Google Scholar
  94. [94]
    Laubenbacher, R.C., Woodburn, C.J.: A new algorithm for the Quillen-Suslin theorem. Beiträge Algebra Geom. 41, 23–31 (2000)MathSciNetzbMATHGoogle Scholar
  95. [95]
    Lenstra, A.K., Lenstra, Jr. H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  96. [96]
    Lequain, Y., Simis, A.: Projective modules over R[X 1, , X n], R a Prüfer domain. J. Pure Appl. Algebra 18(2), 165–171 (1980)Google Scholar
  97. [97]
    Logar, A., Sturmfels, B.: Algorithms for the Quillen-Suslin theorem. J. Algebra 145(1), 231–239 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  98. [98]
    Lombardi, H.: Le contenu constructif d’un principe local-global avec une application à la structure d’un module projectif de type fini. Publications Mathématiques de Besançon. Théorie des nombres (1997)Google Scholar
  99. [99]
    Lombardi, H.: Relecture constructive de la théorie d’Artin-Schreier. Ann. Pure Appl. Logic 91, 59–92 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  100. [100]
    Lombardi, H.: Dimension de Krull, Nullstellensätze et Évaluation dynamique. Math. Z. 242, 23–46 (2002)MathSciNetzbMATHGoogle Scholar
  101. [101]
    Lombardi, H.: Platitude, localisation et anneaux de Prüfer, une approche constructive. Publications Mathématiques de Besançon. Théorie des nombres. Années 1998–2001Google Scholar
  102. [102]
    Lombardi, H.: Hidden constructions in abstract algebra (1) integral dependance relations. J. Pure Appl. Algebra 167, 259–267 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  103. [103]
    Lombardi, H.: Constructions cachées en algèbre abstraite (4) La solution du 17ème problème de Hilbert par la théorie d’Artin-Schreier. Publications Mathématiques de Besançon. Théorie des nombres. Années 1998–2001Google Scholar
  104. [104]
    Lombardi, H.: Constructions cachées en algèbre abstraite (5) Principe local-global de Pfister et variantes. Int. J. Commut. Rings 2(4), 157–176 (2003)MathSciNetzbMATHGoogle Scholar
  105. [105]
    Lombardi, H., Perdry, H.: The Buchberger algorithm as a tool for ideal theory of polynomial rings in constructive mathematics. In: Gröbner Bases and Applications (Proceedings of the Conference 33 Years of Gröbner Bases). Mathematical Society Lecture Notes Series, vol. 251, pp. 393–407. Cambridge University Press, London (1998)Google Scholar
  106. [106]
    Lombardi, H., Quitté, C.: Constructions cachées en algèbre abstraite (2) Le principe local-global. In: Fontana, M., Kabbaj, S.-E., Wiegand, S. (eds.) Commutative Ring Theory and Applications. Lecture Notes in Pure and Applied Mathematics, vol. 131, pp. 461–476. Marcel Dekker, New York (2002)Google Scholar
  107. [107]
    Lombardi, H., Quitté, C.: Seminormal rings (following Thierry Coquand). Theor. Comput. Sci. 392, 113–127 (2008)zbMATHCrossRefGoogle Scholar
  108. [108]
    Lombardi, H., Quitté, C.: Algèbre Commutative. Méthodes Constructives. Modules projectifs de type fini. Cours et exercices. Calvage et Mounet, Paris (2011)zbMATHGoogle Scholar
  109. [109]
    Lombardi, H., Quitté, C.: Commutative Algebra. Constructive Methods. Finite Projective Modules. Springer, New York (2015)Google Scholar
  110. [110]
    Lombardi, H., Yengui, I.: Suslin’s algorithms for reduction of unimodular rows. J. Symb. Comput. 39, 707–717 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  111. [111]
    Lombardi H., Quitté C., Diaz-Toca G. M., Modules sur les anneaux commutatifs. Cours et exercices. Calvage et Mounet, 2014.Google Scholar
  112. [112]
    Lombardi, H., Quitté, C., Yengui, I.: Hidden constructions in abstract algebra (6) The theorem of Maroscia, Brewer and Costa. J. Pure Appl. Algebra 212, 1575–1582 (2008)zbMATHCrossRefGoogle Scholar
  113. [113]
    Lombardi, H., Quitté, C., Yengui, I.: Un algorithme pour le calcul des syzygies sur V[X] dans le cas où V est un domaine de valuation. Commun. Algebra 42(9), 3768–3781 (2014)zbMATHCrossRefGoogle Scholar
  114. [114]
    Lombardi, H., Schuster, P., Yengui, I.: The Gröbner ring conjecture in one variable. Math. Z. 270, 1181–1185 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  115. [115]
    Macaulay2. A Quillen-Suslin package. http://wiki.macaulay2.com/Macaulay2/index.php?title=Quillen-Suslin (2011)
  116. [116]
    Magma (Computational Algebra Group within School of Maths and Statistics of University of Sydney). http://magma.maths.usyd.edu.au/magma (2010)
  117. [117]
    Maroscia, P.: Modules projectifs sur certains anneaux de polynomes. C. R. Acad. Sci. Paris Sér. A 285, 183–185 (1977)MathSciNetzbMATHGoogle Scholar
  118. [118]
    Mialebama Bouesso, A., Sow, D.: Non commutative Gröbner bases over rings. Commun. Algebra 43(2), 541–557 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  119. [119]
    Mialebama Bouesso, A., Valibouze, A., Yengui, I.: Gröbner bases over \(\mathbb{Z}/p^{\alpha }\mathbb{Z}\), \(\mathbb{Z}/m\mathbb{Z}\), \((\mathbb{Z}/p^{\alpha }\mathbb{Z}) \times (\mathbb{Z}/p^{\alpha }\mathbb{Z})\), \(\mathbb{F}_{2}[a,b]/\langle a^{2} - a,b^{2} - b\rangle\), and \(\mathbb{Z}\) as special cases of dynamical Gröbner bases. Preprint (2012)Google Scholar
  120. [120]
    Mines, R., Richman, F., Ruitenburg, W.: A Course in Constructive Algebra. Universitext. Springer, Heidelberg (1988)zbMATHCrossRefGoogle Scholar
  121. [121]
    Mnif, A., Yengui, I.: An algorithm for unimodular completion over Noetherian rings. J. Algebra 316, 483–498 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  122. [122]
    Möller, M., Mora, T.: New constructive methods in classical ideal theory. J. Algebra 100, 138–178 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  123. [123]
    Monceur, S., Yengui, I.: On the leading terms ideals of polynomial ideals over a valuation ring. J. Algebra 351, 382–389 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  124. [124]
    Monceur, S., Yengui, I.: Suslin’s lemma for rings containing an infinite field. Colloq. Math. (in press)Google Scholar
  125. [125]
    Mora, T.: Solving Polynomial Equation Systems I: The Kronecker-Duval Philosophy. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  126. [126]
    Northcott, D.G.: Finite Free Resolutions. Cambridge University Press, Cambridge (1976)zbMATHCrossRefGoogle Scholar
  127. [127]
    Norton, G.H., Salagean, A.: Strong Gröbner bases and cyclic codes over a finite-chain ring. Appl. Algebra Eng. Commun. Comput. 10, 489–506 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  128. [128]
    Norton, G.H., Salagean, A.: Strong Gröbner bases for polynomials over a principal ideal ring. Bull. Aust. Math. Soc. 64, 505–528 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  129. [129]
    Norton, G.H., Salagean, A.: Gröbner bases and products of coefficient rings. Bull. Aust. Math. Soc. 65, 145–152 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  130. [130]
    Norton, G.H., Salagean, A.: Cyclic codes and minimal strong Gröbner bases over a principal ideal ring. Finite Fields Appl. 9, 237–249 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  131. [131]
    Park, H.: A computational theory of Laurent polynomial rings and multidimensional FIR systems. University of Berkeley (1995)Google Scholar
  132. [133]
    Park, H.: Symbolic computations and signal processing. J. Symb. Comput. 37, 209–226 (2004)zbMATHCrossRefGoogle Scholar
  133. [134]
    Park, H.: Generalizations and variations of Quillen-Suslin theorem and their applications. In: Work-shop Grb̈ner Bases in Control Theory and Signal Processing. Special Semester on Grb̈ner Bases and Related Methods 2006, University of Linz, Linz, 19 May 2006Google Scholar
  134. [135]
    Park, H., Woodburn, C.: An algorithmic proof of Suslin’s stability theorem for polynomial rings. J. Algebra 178, 277–298 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  135. [136]
    Pauer, F.: On lucky ideals for Gröbner basis computations. J. Symb. Comput. 14, 471–482 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  136. [137]
    Pauer, F.: Gröbner bases with coefficients in rings. J. Symb. Comput. 42, 1003–1011 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  137. [138]
    Perdry, H.: Lazy bases: a minimalist constructive theory of Noetherian rings. MLQ Math. Log. Q. 54, 70–82 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  138. [139]
    Perdry, H.: Strongly Noetherian rings and constructive ideal theory. J. Symb. Comput. 37, 511–535 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  139. [140]
    Perdry, H., Schuster, P.: Noetherian orders. Math. Struct. Comput. Sci. 24(2), 29 (2014)MathSciNetGoogle Scholar
  140. [141]
    Perdry, H., Schuster, P.: Constructing Gröbner bases for Noetherian rings. Math. Struct. Comput. Sci. 21, 111–124 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  141. [142]
    Pola, E., Yengui, I.: A negative answer to a question about leading terms ideals of polynomial ideals. J. Pure Appl. Algebra 216, 2432–2435 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  142. [143]
    Pola, E., Yengui, I.: Gröbner rings. Acta Sci. Math. (Szeged) 80, 363–372 (2014)Google Scholar
  143. [144]
    Preira, J.-M., Sow, D., Yengui, I.: On Polly Cracker over valuation rings and \(\mathbb{Z}_{n}\). Preprint (2010)Google Scholar
  144. [145]
    Quillen, D.: Projective modules over polynomial rings. Invent. Math. 36, 167–171 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  145. [146]
    Rao, R.A.: The Bass-Quillen conjecture in dimension three but characteristic ≠ 2, 3 via a question of A. Suslin. Invent. Math. 93, 609–618 (1988)Google Scholar
  146. [147]
    Rao, R.A., Swan, R.: A regenerative property of a fibre of invertible alternating polynomial matrices (in preparation)Google Scholar
  147. [148]
    Raynaud, M.: Anneaux Locaux Henséliens. Lectures Notes in Mathematics, vol. 169. Springer, Berlin/Heidelberg/New York (1970)Google Scholar
  148. [149]
    Richman, F.: Constructive aspects of Noetherian rings. Proc. Am. Mat. Soc. 44, 436–441 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  149. [150]
    Richman, F.: Nontrivial use of trivial rings. Proc. Am. Mat. Soc. 103, 1012–1014 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  150. [151]
    Roitman, M.: On projective modules over polynomial rings. J. Algebra 58, 51–63 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  151. [152]
    Roitman, M.: On stably extended projective modules over polynomial rings. Proc. Am. Math. Soc. 97, 585–589 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  152. [153]
    Rosenberg, J.: Algebraic K-Theory and Its Applications. Graduate Texts in Mathematics. Springer, New York (1994)zbMATHCrossRefGoogle Scholar
  153. [154]
    Sasaki, T., Takeshima, T.: A modular method for Gröbner-basis construction over \(\mathbb{Q}\) and solving system of algebraic equations. J. Inform. Process. 12, 371–379 (1989)MathSciNetzbMATHGoogle Scholar
  154. [155]
    Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Math. Ann. 275(1), 105–137 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  155. [156]
    Schreyer, F.-O.: A standard basis approach to Syzygies of canonical curves. J. Reine Angew. Math. 421, 83–123 (1991)MathSciNetzbMATHGoogle Scholar
  156. [157]
    Schuster, P.: Induction in algebra: a first case study. In: 2012 27th Annual ACM/IEEE Symposium on Logic in Computer Science, Proceedings LICS 2012, pp. 581–585. IEEE Computer Society Publications, Dubrovnik (June 2012)Google Scholar
  157. [158]
    Schuster, P.: Induction in algebra: a first case study. Log. Methods Comput. Sci. 9(3), 19 (2013)CrossRefGoogle Scholar
  158. [159]
    Seindeberg, A.: What is Noetherian? Rend. Sem. Mat. Fis. Milano 44, 55–61 (1974)MathSciNetCrossRefGoogle Scholar
  159. [160]
    Serre, J.-P.: Faisceaux algébriques cohérents. Ann. Math. 61, 191–278 (1955)CrossRefGoogle Scholar
  160. [161]
    Serre, J.-P.: Modules projectifs et espaces fibrés à fibre vectorielle. Sém. Dubreil-Pisot, no. 23, Paris (1957/1958)Google Scholar
  161. [162]
    Shekhar, N., Kalla, P., Enescu, F., Gopalakrishnan, S.: Equivalence verification of polynomial datapaths with fixed-size bit-vectors using finite ring algebra. In: ICCAD ’05: Proceedings of the 2005 IEEE/ACM International Conference on Computer-Aided Design, pp. 291–296. IEEE Computer Society, Washington, DC (2005)Google Scholar
  162. [163]
    Simis, A., Vasconcelos, W.: Projective modules over R[X], R a valuation ring are free. Not. Am. Math. Soc. 18(5), 944 (1971)Google Scholar
  163. [164]
    Suslin, A.A.: Projective modules over a polynomial ring are free. Sov. Math. Dokl. 17, 1160–1164 (1976)zbMATHGoogle Scholar
  164. [165]
    Suslin, A.A.: On the structure of the special linear group over polynomial rings. Math. USSR-Izv. 11, 221–238 (1977)zbMATHCrossRefGoogle Scholar
  165. [166]
    Suslin, A.A.: On stably free modules. Mat. Sb. (N.S.), 102(144)(4), 537–550 (1977)Google Scholar
  166. [167]
    Swan, R.: On seminormality. J. Algebra 67, 210–229 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  167. [168]
    Traverso, C.: Seminormality and the Picard group. Ann. Scuola Norm. Sup. Pisa 24, 585–595 (1970)MathSciNetzbMATHGoogle Scholar
  168. [169]
    Traverso, C.: Gröbner Trace Algorithms. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC ’88. Lecture Notes in Computer Science, vol. 358, pp. 125–138 (1988)MathSciNetCrossRefGoogle Scholar
  169. [170]
    Traverso, C.: Hilbert functions and the Buchberger’s algorithm. J. Symb. Comput. 22, 355–376 (1997)MathSciNetCrossRefGoogle Scholar
  170. [171]
    Trinks, W.: Über B. Buchbergers Verfahren, Systeme algebraischer Gleichungen zu lösen. J. Number Theory 10, 475–488 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  171. [172]
    Tolhuizen, L., Hollmann, H., Kalker, A.: On the realizability of Bi-orthogonal M-dimensional 2-band filter banks. IEEE Trans. Signal Process. 43, 640–648 (1995)CrossRefGoogle Scholar
  172. [173]
    Valibouze, A., Yengui, I.: On saturations of ideals in finitely-generated commutative rings and Gröbner rings. Preprint (2013)Google Scholar
  173. [174]
    Vaserstein, L.N.: K 1-theory and the congruence problem. Mat. Zametki 5, 233–244 (1969)MathSciNetGoogle Scholar
  174. [175]
    Vaserstein, L.N.: Operations on orbits of unimodular vectors. J. Algebra 100, 456–461 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  175. [176]
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (2003)Google Scholar
  176. [177]
    Wienand, O.: Algorithms for symbolic computation and their applications. Ph.D. thesis, Kaiserslautern (2011)Google Scholar
  177. [178]
    Winkler, F.: A p-adic approach to the computation of Gröbner bases. J. Symb. Comput. 6, 287–304 (1987)CrossRefGoogle Scholar
  178. [179]
    Yengui, I.: An algorithm for the divisors of monic polynomials over a commutative ring. Math. Nachr. 260, 1–7 (2003)MathSciNetCrossRefGoogle Scholar
  179. [180]
    Yengui, I.: Making the use of maximal ideals constructive. Theor. Comput. Sci. 392, 174–178 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  180. [181]
    Yengui, I.: Dynamical Gröbner bases. J. Algebra 301, 447–458 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  181. [182]
    Yengui, I.: The Hermite ring conjecture in dimension one. J. Algebra 320, 437–441 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  182. [183]
    Yengui, I.: Corrigendum to dynamical Gröbner bases [J. Algebra 301(2), 447–458 (2006)] and to Dynamical Gröbner bases over Dedekind rings [J. Algebra 324(1), 12–24 (2010)]. J. Algebra 339, 370–375 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  183. [184]
    Yengui, I.: Stably free modules over R[X] of rank \(>\dim \mathbf{R}\) are free. Math. Comput. 80, 1093–1098 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  184. [185]
    Yengui, I.: The Gröbner ring conjecture in the lexicographic order case. Math. Z. 276, 261–265 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  185. [186]
    Youla, D.C., Pickel, P.F.: The Quillen-Suslin theorem and the structure of n-dimentional elementary polynomial matrices. IEEE Trans. Circ. Syst. 31, 513–518 (1984)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ihsen Yengui
    • 1
  1. 1.Fac. of Science, Dept. of MathematicsUniversity of SfaxSfaxTunisia

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