A generalization of Gröbner basis algorithms to nilpotent group rings
type: Item
instance of: scholarly article
title: A generalization of Gröbner basis algorithms to nilpotent group rings
zbMATH Open document ID: 0877.16012
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 1997-11-27T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 998034
zbMATH Keywords: right quasicommutative reductions
zbMATH Keywords: sets of generators
zbMATH Keywords: normal forms
zbMATH Keywords: noncommutative generalizations of Gröbner bases methods
zbMATH Keywords: group rings of finitely generated nilpotent groups
zbMATH Keywords: confluent semi-Thue systems
zbMATH Keywords: right ideals
zbMATH Keywords: two-sided ideals in group rings
author: Klaus Madlener
author: Birgit Reinert
reviewed by: U. Kaljulaid
MaRDI profile type: Publication
full work available at URL: s002000050056
OpenAlex ID: W2074122093
DOI: 10.1007/S002000050056
Recommended article: One-sided noncommutative Gröbner bases with applications to computing Green's relations
Recommended article: Q3808248
Recommended article: Solving Polynomial Equation Systems
Recommended article: An introduction to commutative and noncommutative Gröbner bases
Recommended article: Q4394975
Recommended article: Q4761721
Recommended article: The ideal membership problem in non-commutative polynomial rings
Recommended article: Gröbner bases in function rings -- a guide for introducing reduction relations to algebraic structures
Recommended article: Q4394959
Recommended article: Letterplace ideals and non-commutative Gröbner bases.
description: scientific article; zbMATH DE number 998034