## Introducing Reduction to Polycyclic Group Rings - A Comparison of Methods

• 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.

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Author: Birgit Reinert urn:nbn:de:hbz:386-kluedo-4499 Reports on Computer Algebra (ZCA Report) (9) Preprint English 1996 1996 Technische Universität Kaiserslautern 2000/04/03 Gröbner bases ; polycyclic group rings ; rewriting Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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