Computational ideal theory in finitely generated extension rings

Since Buchberger introduced the theory of Grobner bases in 1965 it has become an important tool in constructive algebra and, nowadays, Buchberger's method is fundamental for many algorithms in the theory of polynomial ideals and algebraic geometry. Motivated by the results in polynomial rings a lot of possibilities to generalize the ideas to other types of rings have been investigated. The perhaps most general concept, though it does not cover all possible extensions, is the theory of graded structures due to Robbiano and Mom. But in order to obtain algorithmic solutions for the computation of Grobner 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 Grobner 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 Grobner basis computations. Finally, we discuss which conditions could be changed in order to find further effective Grobner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results. (C) 2000 Elsevier Science B.V. All rights reserved.