A generalization of Gröbner basis algorithms to nilpotent group rings
The rewriting approach to the theory of polynomial ideals in commutative polynomial rings originates from B. Buchberger (1965). He found an algorithm to transform a (finite) set of generators of an ideal in \(k[X]\) into a Gröbner basis of the ideal. This last is a set of polynomials such that every polynomial in the ring has an unique normal form with respect to the reduction that uses basis polynomials as rules; polynomials in the ideal are reduced to zero. The left-hand side of such a rule is defined as the largest monomial in the sense of an admissible term ordering. In this interesting and well-written paper a noncommutative generalization of Gröbner bases methods is given -- to the setting of group rings of finitely generated nilpotent groups. Appropriate `Gröbner bases' are characterized through concepts of `saturation' and `\(s\)-polynomials'. Already in his dissertation (Univ. Kaiserslautern, 1995) \textit{B. Reinert} has developed the approach [SEKI Report SR-93-08, Univ. Kaiserslautern] to use additional information concerning the structure of the group under consideration. Here, using the fact that any finitely generated nilpotent group \(G\) is an extension of a torsion-free nilpotent group by a finite group, the authors present \(G\) by a confluent semi-Thue system of special form. It allows them to define `commutative prefixes' of group elements capturing main features of those in commutative case and through this concept to define `right quasicommutative reduction' for right ideals in \(k[G]\). This approach to find `Gröbner bases' is extended to two-sided ideals in group rings of finitely generated nilpotent groups as well. Earlier, noncommutative extensions of the Buchberger theory have been considered for skew polynomial rings by \textit{V. Weispfenning} [Proc. ISSAC '92, 329-334 (1992)], for free algebras by \textit{F. Mora} [Lect. Notes Comput. Sci. 229, 353-362 (1986; Zbl 0659.16003)] and by \textit{T. Mora} [Theor. Comput. Sci. 134, No. 1, 131-173 (1994; Zbl 0824.68056)], for free group algebras by \textit{A. Rosenmann} [J. Symb. Comput. 16, No. 6, 523-549 (1993; Zbl 0809.16026)], for Weil algebras by \textit{W. Lassner} [EUROCAL '85, Lect. Notes Comput. Sci. 204, 99-115 (1985; Zbl 0632.16003)].
noncommutative generalizations of Gröbner bases methods
group rings of finitely generated nilpotent groups
scientific article; zbMATH DE number 998034