Noncommutative Gr\"obner bases over rings

TL;DR

This paper introduces a method for computing Noncommutative Gr"obner bases over valuation noetherian rings, extending classical algorithms to more general algebraic structures.

Contribution

It generalizes the fundamental theorem on normal forms and adapts Buchberger's algorithm for noncommutative polynomial rings over valuation noetherian rings.

Findings

Generalized the fundamental theorem on normal forms over arbitrary rings.

Extended Buchberger's algorithm to noncommutative polynomial rings over valuation noetherian rings.

Provided a unified approach for computing Gr"obner bases in noncommutative and commutative settings.

Abstract

In this work, it is proposed a method for computing Noncommutative Gr\"obner bases over a valuation n{\oe}therian ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical commutative Gr\"obner bases is generalized for Buchberger's algorithm over $R=\mathcal{V}<x_1,...,x_m>$ a free associative algebra with non-commuting variables, where $\mathcal{V}=\mathbb{Z}/n\mathbb{Z}$ or $\mathcal{V}=\mathbb{Z}$. The process proposed, generalizes previous known technics for the computation of Commutative Gr\"obner bases over a valuation n{\oe}therian ring and/or Noncommutative Gr\"obner bases over a field.