an:01136610
Zbl 0929.16024
Madlener, Klaus; Reinert, Birgit
A generalization of Gr??bner basis algorithms to polycyclic group rings
EN
J. Symb. Comput. 25, No. 1, 23-43 (1998).
00046286
1998
j
16S34 68Q70 20C07 16Z05 13P10
integral group rings; decision problems; membership problem; Gr??bner bases; polycyclic group rings; rewriting systems
Group rings are the subject of extensive studies in mathematics. In 1981 Baumslag, Cannonito and Miller showed that for an integral group ring of a polycyclic group, i.e., a group with a finite subnormal series with cyclic factors, several decision problems including the membership problem for submodules are computable. Studying these ideas Sims described how the connections between special submodule bases enable the membership problem and conventional Gr??bner bases to be solved.
In this paper we present our results which generalize reduction and Gr??bner bases to polycyclic group rings. We want to point out that instead of using the fact that every group ring over a polycyclic group is Noetherian, our approach is oriented towards rewriting which leads to a syntactical characterization of Gr??bner bases in terms of \(s\)-polynomials and a completion-based algorithm with which to compute them.