Computational Ideal Theory in Finitely Generated Extension Rings

One of the most general extensions of Buchberger's theory of Gröbner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Göbner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of finitely generated extension rings and present subclasses of such structures which allow uniform algorithmic solutions of the basic problems in the associated graded ring and, hence, of the computation of Gröbner bases with respect to the graded structure. Among the considered rings there are many of the known generalizations. But, in addition, a wide class of rings appears first time in the context of algorithmic Gröbner basis computations. Finally, we discuss which conditions could be changed in order to find further effective Gröbner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results