Gröbner bases and Buchberger's algorithm have been generalized to monoid and group rings. In this paper we summarize procedures from this field and present a description of their implementation in the system Mrc V 1.0.
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 from Baumslag, Cannonito and Miller and studied by Sims is outlined. We want to analyze, how thesedecision methods are related to Gröbner bases. Therefore, we define effective reduction for group rings over Abelian groups, nilpotent groups and moregeneral polycyclic groups. Using these reductions we present generalizations of Buchberger's Gröbner basis method by giving an appropriate definition of"Gröbner bases" in the respective setting and by characterizing them using concepts of saturation and s-polynomials.