Gröbner Bases Property on Elimination Ideal in the Noncommutative Case

Summary

Abstract

Some generalizations to noncommutative algebras of the Gröbner bases property on elimination ideal are discussed here, together with their application to determine Gröbner presentations, independent polynomials modulo an ideal, algebra membership, and algebra equality. In addition, it is also shown how it may be possible to compute the normal closure of a subgroup by means of Gröbner presentations.

Introduction

The notion of Gröbner basis and related techniques are remarkable contributions to the solution of problems by algorithmic way in polynomial ideal theory. The greatest number of applications of the Gröbner bases method have been, for the time being, in commutative polynomial algebras. In Mora F. (1986), the concept of Gröbner basis has been generalized to noncommutative free algebras, allowing to extend the field of applications. Many Gröbner bases results, for the commutative case, are based on the well known Elimination Ideal Property of these bases; Buchberger B. (1987), Gianni P. et al. (1988), Shannon D. et al. (1988), Kalkbrener M. (1991), are some examples. In this paper we show how this property can be gene- ralized to free associative K-algebras and, by means of some applications, illustrate that this generalized property could become similarly applicable as its starting point in commutative polynomial rings.