Non-commutative Gröbner bases in algebras of solvable type. (English) Zbl 0715.16010

Let \(K\) be a commutative field, let \(X=\{X_ 1,...,X_ n\}\) be a set of commuting indeterminates, let \(T\) denote the free commutative monoid over \(X\) (i.e. the set of power-products in the indeterminates of \(X\)) and let \(<\) be a fixed order on \(T\) which makes it a fully ordered monoid. Let us denote \(R\) the set of polynomials over \(K\) in the indeterminates of \(X\). A non-commutative ring of solvable type is obtained from \(R\) by equipping it with a ring product * which satisfies some compatibility conditions with the product by elements of \(K\) or \(T\) and such that for every \(i,j\in [[ 1,n]]\), there exists a non-zero constant \(c_{i,j}\in K\) and an element \(p_{i,j}\) of \(R\) such that \(X_ i*X_ j=c_{i,j}X_ iX_ j+p_{i,j}\).
The authors prove first some basic properties of these non-commutative rings and show that iterated Ore differential extensions, quotients of free associative algebras by general kinds of commutation relations and enveloping algebras of finite dimensional Lie algebras are non-commutative rings of solvable type. Then they study several properties and characterizations of left, right and two-sided Gröbner bases in their framework. They present in particular some algorithms for computing products and Gröbner bases in polynomial rings of solvable type. Finally they show that the word problem and the ideal-membership problem are solvable in algebras of solvable type (i.e. quotients of solvable polynomial rings).
Reviewer: D.Krob


16S36 Ordinary and skew polynomial rings and semigroup rings
68W30 Symbolic computation and algebraic computation
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
16S32 Rings of differential operators (associative algebraic aspects)
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16-04 Software, source code, etc. for problems pertaining to associative rings and algebras
03D40 Word problems, etc. in computability and recursion theory
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[1] Apel, J.; Lassner, W., An extension of Buchberger’s algorithm and calculations in enveloping fields of Lie Algebras, J. Symbolic Comp., 6, 361-370 (1988) · Zbl 0663.68044
[2] Bergmann, G. M., The diamond lemma in ring theory, Adv. Math., 29, 178-218 (1978)
[3] Boone, W. W., The word problem, Ann. Math., 70, 207-265 (1959) · Zbl 0102.00902
[4] Buchberger, B., Some properties of Gröbner bases for polynomial ideals, ACM SIGSAM Bull., 10, 4, 19-24 (1976)
[5] Buchberger, B., Gröbner bases: an algorithmic method in polynomial ideal theory, (Bose, N. K., Recent Trends in Multidimensional Systems (1985), Reidel: Reidel Dordrecht), chap. 6 · Zbl 0587.13009
[6] Dickson, L. E., Finiteness of the odd perfect and primitive abundant numbers with \(n\) distinct prime factors, Amer. J. Math., 35, 413-426 (1913) · JFM 44.0220.02
[7] Dixmier, J., Algèbres enveloppantes (1974), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0308.17007
[8] Eilenberg, S.; Schutzenberger, M. F., Rational sets in commutative monoids, J. Algebra, 13, 173-191 (1969) · Zbl 0206.02703
[9] El From, Y., Sur les algèbres de type résoluble, (Thèse de 3e cycle (1983), Univ. Paris 6) · Zbl 0880.16012
[10] Galligo, A., Some algorithmic questions on ideals of differential operators, (Proc. EUROCAL ’85, Springer LNCS, 204 (1985)), 413-421 · Zbl 0634.16001
[11] Gebauer, R.; Kredel, H., (Distributive Polynomial System. Several Technical Reports (1983), University of Heidelberg: University of Heidelberg Heiddlberg, FRG)
[12] Huet, G., Confluent reductions: Abstract properties and applications to term rewriting systems, J. ACM, 27, 797-821 (1980) · Zbl 0458.68007
[13] Jacobson, N., (Lie Algebras (1962), Dover: Dover New York) · Zbl 0121.27504
[14] Kandri-Rody, A.; Kapur, D., Algorithms for computing the Gröbner bases of polynomial ideals over various Euclidean rings, (Proc. EUROSAM ’84, Springer LNCS, 174 (1984)), 195-205 · Zbl 0564.13001
[15] Kandri-Rody, A.; Kapur, D., An algorithm for computing the Gröbner basis of a polynomial ideal over a Euclidean ring, J. Symb. Comp., 6, 37-58 (1988) · Zbl 0658.13016
[16] Lassner, W., Symbol representation of non-commutative algebras, (Proc. EUROCAL ’85, Springer LNCS, 204 (1985)), 99-115
[17] Lesieur, L., Conditions Noéthériennes dans l’anneau de polynomes de Ore A[X, σ, δ], Séminaire d’algèbre P. Dubreil, Paris 1977-78, Springer LNM, 641, 220-234 (1978)
[18] Mayr, E. W.; Meyer, A. R., The complexity of the word problem for commutative semigroups and polynomial ideals, Adv. Math., 46, 305-329 (1982) · Zbl 0506.03007
[19] Mora, F., Gröbner bases for non-commutative polynomial rings, (Proc. AAECC-3, Springer LNCS, 229 (1986)), 353-362 · Zbl 0659.16003
[20] Ore, O., Theory of non-commutative polynomials, Ann. Math., 34, 480-508 (1933) · JFM 59.0925.01
[21] van der Waerden, B. L., (Algebra (1967), Springer: Springer Berlin)
[22] Weispfenning, V., Constructing universal Gröbner bases, (Proc AAECC-5, LNCS, 356 (1988), Springer: Springer Berlin), 408-417
[23] Winkler, F.; Buchberger, B.; Lichtenberger, F.; Rolletschek, H., An algorithm for constructing canonical bases of polynomial ideals, ACM/TOMS, 11, 66-78 (1985) · Zbl 0575.68047
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