**Klaus Madlener, Birgit Reinert
Universität Kaiserslautern
67663 Kaiserslautern
{madlener,reinert}@informatik.uni-kl.de
presented at the
Workshop on Symbolic Rewriting Systems
Monte Verita, May 1995**

**October, 1997**

The concept of algebraic simplification is of great importance for the field
of symbolic computation in computer algebra.
In this paper we review some fundamental concepts concerning reduction rings
in the spirit of Buchberger.
The most important properties of reduction rings are presented.
The techniques for presenting monoids or groups by string rewriting systems
are used to define several types of reduction in monoid and group rings.
Gröbner bases in this setting arise naturally as generalizations of the
corresponding known notions in the commutative and some non-commutative cases.
Several results on the connection of the word problem and the
congruence problem are proven.
The concepts of saturation and completion are introduced for monoid
rings having a finite convergent presentation by a semi-Thue system.
For certain presentations, including free groups and
context-free groups, the existence of finite Gröbner bases for
finitely generated right ideals is shown and a procedure to compute
them is given.

- 1. Introduction
- 2. Fundamental Relations Between Semi-Thue Systems and Monoid Rings
- 3. Defining Reduction in
- 4. Group Rings
- 5. Conclusions
- Bibliography
- 6. Appendix
- About this document ...