22 - Gröbner Bases in Non-Commutative Reduction Rings

Summary

Abstract

Gröbner bases of ideals in polynomial rings can be characterized by properties of reduction relations associated with ideal bases. Hence reduction rings can be seen as rings with reduction relations associated to subsets of the ring such that every finitely generated ideal has a finite Gröbner basis. This paper gives an axiomatic framework for studying reduction rings including non-commutative rings and explores when and how the property of being a reduction rings is preserved by standard ring constructions such as quotients and sums of reduction rings, and polynomial and monoid rings over reduction rings.

Introduction

Reasoning and computing in finitely presented algebraic structures is wide-spread in many fields in mathematics, physics and computer science. Reduction in the sense of simplification combined with appropriate completion methods is one general technique which is often successfully applied in this context, e.g. to solve the word problem and hence to compute effectively in the structure.

One fundamental application of this technique to polynomial rings was provided by B. Buchberger (1965) in his uniform effective solution of the ideal membership problem establishing the theory of Gröbner bases. These bases can be characterized in various manners, e.g. by properties of a reduction relation associated with polynomials (confluence or all elements in the ideal reduce to zero) or by special representations for the ideal elements with respect to a Gröbner basis (standard representations).