26 - On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings

Summary

Abstract

In this paper we generalize some basic applications of Gröbner bases in commutative polynomial rings to the non-commutative case. We define a non-commutative elimination order. Methods of finding the intersection of two ideals are given. If both the ideals are monomial we deduce a finitely written basis for their intersection. We find the kernel of a homomorphism, and decide membership of the image. Finally we show how to obtain a Gröbner basis for an ideal by considering a related homogeneous ideal.

The method of Gröbner bases, introduced by Bruno Buchberger in his thesis (1965), have become a powerful tool for constructive problems in polynomial ideal theory and related domains. Generalizations of the basic ideas to the non-commutative setting was done, as an theoretical instrument, by Bokut (1976) and Bergman (1978). From the constructive point of view, the non-commutative version of Buchberger's algorithm was presented by Mora (1986). For some special classes of non-commutative rings, Gröbner bases has been studied in more detail, e.g. solvable algebras by Kandri-Rody and Weispfenning (1990).

As the title indicates, we will here consider Gröbner bases in non-commutative polynomial rings, i.e. free associative algebras (over some field). Most of the results are just easy generalizations of the theory of Gröbner basis in commutative polynomial rings, which can be found e.g. in the textbook by Adams and Loustaunau (1994), or in the original paper by Buchberger (1985).