Noncommutative Gröbner Bases over Rings

Abstract

In this work, we propose a method for computing noncommutative Gröbner bases over a noetherian valuation ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical commutative Gröbner bases is generalized to Buchberger's algorithm over R = 𝒱 ⟨ x ₁,…, x _m  ⟩, a free associative algebra with noncommuting variables, where 𝒱 = ℤ/nℤ and 𝒱 = ℤ.

The proposed process generalizes previous known techniques for the computation of commutative Gröbner bases over a nætherian valuation ring and/or noncommutative Gröbner bases over a field.

Key Words:

• Buchberger algorithm
• Commutative Gröbner bases
• Dikson lemma
• Noncommutative Gröbner bases
• Termination theorem
• Valuation nætherian ring

2010 Mathematics Subject Classification:

• 13P10
• 12Y05
• 16Z05

ACKNOWLEDGMENT

The first author is grateful to the IMU Berlin Einstein Foundation, the Berlin Mathematical School, and to Pr. Dr. Klaus Altmann for receiving him in Berlin during the writing of a part of this paper.

Notes

Communicated by A. Elduque.