Book contents
- Frontmatter
- Contents
- Preface
- Programme Committee
- Tutorials
- Research Papers
- The Fractal Walk
- Gröbner Bases Property on Elimination Ideal in the Noncommutative Case
- 17 The CoCoA 3 Framework for a Family of Buchberger-like Algorithms
- 18 Newton Identities in the Multivariate Case: Pham Systems
- 19 Gröbner Bases in Rings of Differential Operators
- 20 Canonical Curves and the Petri Scheme
- 21 The Buchberger Algorithm as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics
- 22 Gröbner Bases in Non-Commutative Reduction Rings
- 23 Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field
- 24 De Nugis Groebnerialium 1: Eagon, Northcott, Gröbner
- 25 An application of Gröbner Bases to the Decomposition of Rational Mappings
- 26 On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings
- 27 Full Factorial Designs and Distracted Fractions
- 28 Polynomial interpolation of Minimal Degree and Gröbner Bases
- 29 Inversion of Birational Maps with Gröbner Bases
- 30 Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated
- 31 Parallel Computation and Gröbner Bases: An Application for Converting Bases with the Gröbner Walk
- Appendix An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (translated by Michael Abramson and Robert Lumbert)
- Index of Tutorials
26 - On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Preface
- Programme Committee
- Tutorials
- Research Papers
- The Fractal Walk
- Gröbner Bases Property on Elimination Ideal in the Noncommutative Case
- 17 The CoCoA 3 Framework for a Family of Buchberger-like Algorithms
- 18 Newton Identities in the Multivariate Case: Pham Systems
- 19 Gröbner Bases in Rings of Differential Operators
- 20 Canonical Curves and the Petri Scheme
- 21 The Buchberger Algorithm as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics
- 22 Gröbner Bases in Non-Commutative Reduction Rings
- 23 Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field
- 24 De Nugis Groebnerialium 1: Eagon, Northcott, Gröbner
- 25 An application of Gröbner Bases to the Decomposition of Rational Mappings
- 26 On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings
- 27 Full Factorial Designs and Distracted Fractions
- 28 Polynomial interpolation of Minimal Degree and Gröbner Bases
- 29 Inversion of Birational Maps with Gröbner Bases
- 30 Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated
- 31 Parallel Computation and Gröbner Bases: An Application for Converting Bases with the Gröbner Walk
- Appendix An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (translated by Michael Abramson and Robert Lumbert)
- Index of Tutorials
Summary
Abstract
In this paper we generalize some basic applications of Gröbner bases in commutative polynomial rings to the non-commutative case. We define a non-commutative elimination order. Methods of finding the intersection of two ideals are given. If both the ideals are monomial we deduce a finitely written basis for their intersection. We find the kernel of a homomorphism, and decide membership of the image. Finally we show how to obtain a Gröbner basis for an ideal by considering a related homogeneous ideal.
The method of Gröbner bases, introduced by Bruno Buchberger in his thesis (1965), have become a powerful tool for constructive problems in polynomial ideal theory and related domains. Generalizations of the basic ideas to the non-commutative setting was done, as an theoretical instrument, by Bokut (1976) and Bergman (1978). From the constructive point of view, the non-commutative version of Buchberger's algorithm was presented by Mora (1986). For some special classes of non-commutative rings, Gröbner bases has been studied in more detail, e.g. solvable algebras by Kandri-Rody and Weispfenning (1990).
As the title indicates, we will here consider Gröbner bases in non-commutative polynomial rings, i.e. free associative algebras (over some field). Most of the results are just easy generalizations of the theory of Gröbner basis in commutative polynomial rings, which can be found e.g. in the textbook by Adams and Loustaunau (1994), or in the original paper by Buchberger (1985).
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- Gröbner Bases and Applications , pp. 463 - 472Publisher: Cambridge University PressPrint publication year: 1998
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