Noncommutative Gr\"obner bases over rings

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 $R=\mathcal{V}<x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_m>$ a free associative algebra with non-commuting variables, where $\mathcal{V}=\mathbb{Z}\mathrm{/}n\mathbb{Z}$ 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.