On Some Basic Applications of Grr Obner Bases in Non-commutative Polynomial Rings

In this paper we generalize some basic applications of Grr obner bases in commutative polynomial rings to the non-commutative case. We deene a non-commutative elimination order. Methods of nding the intersection of two ideals are given. If both the ideals are monomial we deduce a nitely written basis for their intersection. We nd the kernel of a homomorphism, and decide membership of the image. Finally we show how to obtain a Grr obner basis for an ideal by considering a related homogeneous ideal. The method of Grr obner 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, Grr obner 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 Grr obner bases in non-commutative polynomial rings, i.e. free associative algebras (over some eld). Most of the results are just easy generalizations of the theory of Grr obner basis in commu-tative polynomial rings, which can be found e.g. in the textbook by Adams and Loustaunau (1994), or in the original paper by Buchberger (1985). In section 2 we deene an order allowing us to use elimination techniques. Section 3 concerns intersection of ideals. We nd a Grr obner basis for the intersection of two ideals. In the special case when both the ideals are mono-mial, we show how automata can be used to obtain a nitely written basis. The material in section 4, where we nd a Grr obner basis for the kernel of a homomorphism, and decide membership of the image, is nearly identical 1