A critical-pair/completion algorithm for finitely generated ideals in rings

  • B. Buchberger
Section II: Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 171)


In 1965, the author introduced a "critical-pair/completion" algorithm that starts from a finite set F of polynomials in K[x1,...,xn] (K a field) and produces a set G of polynomials such that the ideals generated by F and G are identical, but G is in a certain standard form (G is a "Gröbner-basis"), for which a number of important decision and computability problems in polynomial ideal theory can be solved elegantly. In this paper, it is shown how the critical-pair/completion approach can be extended to general rings. One of the difficulties lies in the fact that, in general, the generators of an ideal in a ring do not naturally decompose into a "head" and a "rest" (left-hand side and right-hand side). Thus, the crucial notions of "reduction" and "critical pair" must be formulated in a new way that does not depend on any "rewrite" nature of the generators. The solution of this problem is the starting point of the paper. Furthermore, a set of reduction axioms is given, under which the correctness of the algorithm can be proven and which are preserved when passing from a ring R to the polynomial ring R[x1,...,,xn]. Z[x1,...,xn] is an important example of a ring in which the critical-pair/completion approach is possible.


Reduction Relation Polynomial Ring Polynomial Ideal Residue Class Commutative Semigroup 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • B. Buchberger
    • 1
  1. 1.Mathematisches InstitutJohannes Kepler UniversitätLinzAustria

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