A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings

@article{Madlener1998AGO,
  title={A Generalization of Gr{\"o}bner Basis Algorithms to Polycyclic Group Rings},
  author={Klaus Madlener and Birgit Reinert},
  journal={J. Symb. Comput.},
  year={1998},
  volume={25},
  pages={23-43}
}
It is well-known that for the integral group ring of a polycyclic group several decision problems are decidable, in particular the ideal membership problem. In this paper we define an effective reduction relation for group rings over polycyclic groups. This reduction is based on left multiplication and hence corresponds to left ideals. Using this reduction we present a generalization of Buchberger's Grobner basis method by giving an appropriate definition of “Grobner bases” in this setting and… 

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References

SHOWING 1-10 OF 19 REFERENCES

A Generalization of Gröbner Basis Algorithms to Nilpotent Group Rings

This paper defines an effective reduction for group rings over finitely generated nilpotent groups — a subclass of polycyclic-by-finite groups — and presents a generalization of Buchberger’s Gröbner basis method.

Introducing Reduction to Polycyclic Group Rings - A Comparison of Methods

t is well-known that for the integral group ring of a polycyclic group several decision problems are decidable. In this paper a technique to solve themembership problem for right ideals originating

On Gröbner bases in monoid and group rings

A completion procedure for finitely generated right ideals in Z[H] is given, where H is an ordered monoid presented by a finite, convergent semi - Thue system (Sigma; T), and termination can be shown.

An Algorithm for Constructing Gröbner and Free Schreier Bases in Free Group Algebras

An algorithm for computing Grobner and free canonical Schreier bases for finitely generated one-sided ideals in free group algebras and obtains an algorithm similar to the Nielsen-Hall algorithm for constructing free bases for subgroups of free groups.

Computing Gröbner bases in monoid and group rings

This paper summarizes procedures from this field and presents a description of their implementation in the system Mrc V 1.0.

Non-Commutative Gröbner Bases in Algebras of Solvable Type

Finite Gröbner bases in non-Noetherian skew polynomial rings

One of the widest classes of lloll–collllll~~tati~~e 1~”--algebras known at present, where every (fhlitely generated) ideal has a finite Grobner basis, is the class of solvable polynomial rings studied in [KRW].

An Extension of Buchberger's Algorithm and Calculations in Enveloping Fields of Lie Algebras

An Introduction to Commutative and Noncommutative Gröbner Bases

  • T. Mora
  • Mathematics
    Theor. Comput. Sci.
  • 1994

Computable algebra and group embeddings