Abstract:
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 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. The approach is extended to two-sided ideals and a discussion on a Gröbner bases approach for right ideals is included.

Klaus Madlener, Birgit Reinert
Fachbereich Informatik
Postfach 3049
67653 Kaiserslautern, Germany