Noncommutative Gröbner bases over rings

Abstract

In this work, it is proposed a method for computing Noncommutative Gröbner bases over a valuation nœtherian ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical commutative Gröbner 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 Gröbner bases over a valuation nœtherian ring and/or Noncommutative Gröbner bases over a field.