Title | **A Generalization of Gröbner Basis Algorithms to Nilpotent Group Rings** |

Author(s) | Klaus Madlener, Birgit Reinert |

Type | Article in Journal |

Abstract | It is well-known that for the integral group ring of a polycyclic-by-finite group several decision problems including the membership problem for right ideals are decidable. In this paper we define an effective reduction for group rings over finitely generated nilpotent groups -- a subclass of polycyclic-by-finite groups. Using this reduction we present a generalization of Buchberger's Gröbner basis method by giving an appropriate definition of "Gröbner bases" in this setting and by characterizing them using the concepts of saturation and s-polynomials. Our approach allows to compute such Gröbner bases by completion based algorithms and to use these bases to solve the membership problem for right and two-sided ideals in finitely generated nilpotent group rings using Gröbner basis algorithms and reduction. |

Keywords | Gröbner bases, Nilpotent group rings, Rewriting |

ISSN | 0938-1279 |

Language | English |

Journal | Applicable Algebra in Engineering, Communication and Computing |

Volume | 8 |

Number | 2 |

Pages | 103-123 |

Publisher | Springer-Verlag GmbH |

Year | 1997 |

Month | January |

Translation |
No |

Refereed |
No |

Organization |
Universität Kaiserslautern |