Gröbner Bases Theory and The Diamond Lemma

Abstract

Commutative Gröbner bases theory is well known and widely used. In this thesis, we will discuss thoroughly its generalization to noncommutative polynomial ring <em>k</em><<em>X</em>> which is also an associative free algebra. We introduce some results on monomial orders due to John Lawrence and the author. We show that a noncommutative monomial order is a well order while a one-sided noncommutative monomial order may not be. Then we discuss the generalization of polynomial reductions, S-polynomials and the characterizations of noncommutative Gröbner bases. Some results due to Mora are also discussed, such as the generalized Buchberger's algorithm and the solvability of ideal membership problem for homogeneous ideals. At last, we introduce Newman's diamond lemma and Bergman's diamond lemma and show their relations with Gröbner bases theory.