A Generalization of Gröbner Basis Algorithms to Nilpotent Group Rings

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.