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**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).

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

### MSC:

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 |

### Keywords:

free commutative monoids; iterated Ore differential extensions; free associative algebras; commutation relations; enveloping algebras; non-commutative rings of solvable type; left, right and two-sided Gröbner bases; algorithms; polynomial rings of solvable type; word problem; ideal-membership problem; algebras of solvable type
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\textit{A. Kandri-Rody} and \textit{V. Weispfenning}, J. Symb. Comput. 9, No. 1, 1--26 (1990; Zbl 0715.16010)

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