Groebner Rings and Modules
Bruno Buchberger
In: Proceedings of SYNASC 2001 (The 3rd International Workshop on Symbolic and Numeric
Algorithms for Scientific Computing), Oct. 2-5, 2001, Timisoara, Romania, pp.22-25.
(Copyright: University of the West at Timisoara.)
ABSTRACT:
We sketch an axiomatic approach for the theory of Groebner bases in rings
and modules.
A Groebner ring is a ring with three additional operations: a Noetherian ordering, a
ring quotient, and an operation called "least common reducible". In an earlier paper
(1985) we had introduced axioms for slightly more complicated additional operations
and we pose the problem of finding appropriate axioms for the above three operations
in order to guarantee that
- a ring satisfying the axioms allow the construction of Groebner bases by considering
finitely many least common reducibles and
- the axioms are preserved if one goes from a ring to the polynomial ring over the given
ring and to various other rings that can be constructed from the given ring by various
constructive functors.