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Introducing Reduction to Polycyclic Group Rings -
A Comparison of Methods
Birgit Reinert1
Fachbereich Informatik
Universität Kaiserslautern
67663 Kaiserslautern
Germany
[email protected]
October 1996
Abstract:
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
Gröbner bases.
Therefore, we define effective reduction for group rings over
Abelian groups, nilpotent groups and more general 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.
Keywords: Gröbner bases, polycyclic group rings, rewriting
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