ABSTRACT

Computations have always been and will always be an integral part of mathematics. In abstract ring theory and module theory, the introduction of Gröbner bases, adds a new and powerful tool to our arsenal of computational techniques. Why are Gröbner bases important? First they allow the development of algorithms to study the structure of rings and of modules over the rings. Some of these algorithms are already being employed computer algebra packages such as Mathematica, Maple, Macauley, CoCoa and Axiom (for commutative algebra) and our own package, Groebner (for noncommutative algebra). The utility of being able to compute difficult examples which previously were not accessible to pencil and paper computations 168is evident. A second reason for the study of these ideas is that they introduce new and interesting problems into ring and module theory. It is not uncommon to see some computer output and be led to mathematical questions or conjectures inspired by the output. Patterns, not readily visible by hand calculations, can be seen by the study of the computer output.