Gröbner bases for modules

  • José Bueso
  • José Gómez-Torrecillas
  • Alain Verschoren
Part of the Mathematical Modelling book series (MMTA, volume 17)

Abstract

Until now, we have only dealt with ideals of PBW rings and shown how Gröbner bases may efficiently be used to solve classical questions like the membership or equality problem. In the present chapter we will generalize this approach and show how it applies as well to submodules of the free module R n Actually, it appears that this set-up may also be used to calculate syzygy modules, both in the ideal and (more generally) the module case. The introduction and study of Gröbner bases themselves depends heavily upon the use of admissible orders on \({{\Bbb N}^{n,\left( m \right)}} = {{\Bbb N}^n} \times \left\{ {1, \ldots ,m} \right\}\) a notion which generalizes the homonymous one over ℕ n and which possesses similar properties. On the other hand, it should also be clear that an analogue of the division algorithm for modules will play a leading role throughout.

Keywords

Left Ideal Finite Rank Division Algorithm Newton Diagram Admissible Order 
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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Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • José Bueso
    • 1
  • José Gómez-Torrecillas
    • 1
  • Alain Verschoren
    • 2
  1. 1.University of GranadaGranadaSpain
  2. 2.University of AntwerpAntwerpBelgium

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