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

## Abstract

In 1965, the author introduced a "critical-pair/completion" algorithm that starts from a finite set F of polynomials in K[x_{1},...,x_{n}] (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[x_{1},...,,x_{n}]. Z[x_{1},...,x_{n}] is an important example of a ring in which the critical-pair/completion approach is possible.

## Keywords

Reduction Relation Polynomial Ring Polynomial Ideal Residue Class Commutative Semigroup## Preview

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## References

- Ayoub, C. W., 81: On Constructing Bases for Ideals in Polynomial Rings over the Integers. The Pennsylvania State University, Dept. of Mathematics, Report Nr. 8184, 1981, submitted to publication.Google Scholar
- Ballantyne, A. M., Lankford, D. S., 81: New Decision Algorithms for Finitely Presented Commutative Semigroups. Comp. and Maths. with Appls. 7 (1981), 159–165.Google Scholar
- Bauer, G., 81: The Representation of Monoids by Confluent Rule Systems. University of Kaiserlautern, FRG, Fachbereich Informatik, Ph.D. Thesis, 1981.Google Scholar
- Bergman, G. M., 78: The Diamond Lemma for Ring Theory. Adv. Math. 29, (1978), 178–218.Google Scholar
- Buchberger, B., 65: An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal (German). Univ. of Innsbruck, Austria, Math. Inst., Ph.D. Thesis, 1965.Google Scholar
- Buchberger, B., 70: An Algorithmical Criterion for the Sovability of Algebraic Systems of Equations (German). Aequationes mathematicae 4/3 (1970), 374–383.Google Scholar
- Buchberger, B., 76: A Theoretical Basis for the Reduction of Polynomials to Canonical Form. ACM SIGSAM Bull. 10/3 (1976), 19–29.Google Scholar
- Buchberger, B., 76a: Some Properties of Gröbner Bases for Polynomial Ideals. ACM SIGSAM Bull. 10/4 (1976), 19–24.Google Scholar
- Buchberger, B., 79: A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner-Bases. Proc. EUROSAM 79, Marseille (Ng, W., ed.), Lecture Notes in Computer Science 72 (1979), 3–21.Google Scholar
- Buchberger, B., 83: A Note on the Complexity of Constructing Gröbner-Bases. Proc. of the EUROCAL 83 Conf., London, Lecture Notes in Computer Science, Sringer, 1983, to appear.Google Scholar
- Buchberger, B., 83a: A Critical-Pair/Completion Algorithm in Reduction Rings. University of Linz, Austria, Math. Institute, Technical Report Nr. CAMP 83-21.0, 1983.Google Scholar
- Buchberger, B., Loos, R., 82: Algebraic Simplification. In: Computer Algebra (B. Buchberger, G. Collins, R. Loos eds.), Springer, Wien New York, 1982, 11–43.Google Scholar
- Guiver, J. P., 82: Contributions to Two-Dimensional System Theory. Univ. of Pittsburgh, Math. Dept., Ph.D. Thesis, 1982.Google Scholar
- Hurd, C. B., 70: Concerning Ideals in Z[x] and Z
_{p}n[x]. The Pennsylvania State University, Department of Mathematics, Ph.D. Thesis, 1970.Google Scholar - Knuth, D. E., Bendix, P. B., 67: Simple Word Problems in Universal Algebras. Proc. of the Conf. on Computational Problems in Abstract Algebra, Oxford, 1967, (Leech, J., ed.), Oxford: Pergamon Press, 1970.Google Scholar
- Kronecker, L., Hensel, K., 01: Lectures on Number Theory (German). Leipzig, 1901.Google Scholar
- Lauer, M., 76: Canonical Representatives for the Residue Classes of a Polynomial Ideal. University of Kaiserslautern, FRG, Dept. of Mathematics, Diploma Thesis, 1976.Google Scholar
- Llopis de Trias, R., 83: Canonical Forms for Residue Classes of Polynomial Ideals and Term Rewriting Systems. Univ. Aut. de Madrid, Division de Mathematicas, submitted to publication, 1983.Google Scholar
- Redei, L., 56: Equivalence of the Theorems of Kronecker-Hensel and Szekeres (German). Acta Sci. Math. Szeged 17 (1956), 198–202.Google Scholar
- Richman, F., 74: Constructive Aspects of Noetherian Rings. Proc. AMS 44/2 (1974), 436–441.Google Scholar
- Schaller, S., 79: Algorithmic Aspects of Polynomial Residue Class Rings. University of Wisconsin, Madison, Ph.D. Thesis, Comput. Sci. Tech. Rep. 370, 1979.Google Scholar
- Simmons, H., 70: The Solution of a Decision Problem for Several Classes of Rings. Pac. J. Math. 34 (1970), 547–557.Google Scholar
- Sims, C., 78: The Role of Algorithms in the Teaching of Algebra. In: Topics in Algebra (Newman, M. F. ed.), Lecture Notes in Mathematics 697 (1978), Springer, 95–107.Google Scholar
- Spear, D., 77: A Constructive Approach to Commutative Ring Theory. Proc. of the MACSYMA Users' Conference, Berkeley, 1977, (Fateman, R.J., ed.), published by MIT, 369–376.Google Scholar
- Szekeres, G., 52: A Canonical Basis for the Ideals of a Polynomial Domain. Am. Math. Monthly 59 (1952), 379–386.Google Scholar
- Szekeres, G., 65: Metabelian Groups with Two Generators. Proc. Internat. Conf. Theory of Groups (Canberra 1965), Gordon & Breach, 1967, 323–346.Google Scholar
- Szekeres, G., 75: Homogeneous Ideals in K[x,y,z]. Acta Math. Acad. Sci. Hungar. 26 (1975), 355–367.Google Scholar
- Trinks, W., 78: On B. Buchberger's method for Solving Algebraic Equations (German). J. Number Theory 10/4 (1978), 475–488 (preprint 1977).Google Scholar
- Trotter, P. G., 69: A Canonical Basis for Ideals of Polynomials in Several Variables and With Integer Coefficients. Univ. of New South Wales, Ph.D. Thesis, 1969.Google Scholar
- Trotter, P. G., 78: Ideals in Z[x,y]. Acta Math. Acad. Sci. Hungar. 32 (1–2) (1978), 63–73.Google Scholar
- Winkler, F., Buchberger, B., 83: A Criterion for Eliminating Unnecessary Reductions in the Knuth-Bendix Algorithm. Coll. on Algebra, Combinatorics and Logic in Computer Science, Györ, Sept. 12–16, 1983, 21 p.Google Scholar
- Zacharias, G., 78: Generalized Gröbner Bases in Commutative Polynomial Rings. MIT, Dept. Comput. Sci., Bachelor Thesis, 1978.Google Scholar