String Rewriting and Grr Obner Bases { a General Approach to Monoid and Group Rings Presented at the Workshop on Symbolic Rewriting Systems

The concept of algebraic simpliication is of great importance for the eld 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 deene several types of reduction in monoid and group rings. Grr obner 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 nite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, the existence of nite Grr obner bases for nitely generated right ideals is shown and a procedure to compute them is given.