Introducing Reduction to Polycyclic Group Rings - A Comparison of Methods

It 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 the membership problem for right ideals originating from Baumslag, Cannonito and Miller and studied by Sims is outlined. We want to analyze, how these decision methods are related to Grr obner bases. Therefore, we deene eeective reduction for group rings over Abelian groups, nilpotent groups and more general polycyclic groups. Using these reductions we present generalizations of Buchberger's Grr obner basis method by giving an appropriate deenition of \Grr obner bases" in the respective setting and by characterizing them using concepts of saturation and s-polynomials.