Computational ideal theory in finitely generated extension rings

One of the most general extensions of Buchberger's theory of Grobner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Gr obner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of nitely 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 Gr obner 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 rst time in the context of algorithmic Grobner basis computations. Finally, we discuss which conditions could be changed in order to nd further e ective Gr obner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results.