Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra

In this paper we introduce an algebra embedding ι : K<X> → S from the free associative algebra K<X> generated by a finite or countable set X into the skew monoid ring S = P ∗ Σ defined by the commutative polynomial ring P = K [X × N^∗] and by the monoid Σ = <σ> generated by a suitable endomorphism σ : P → P. If P = K[X] is any ring of polynomials in a countable set of commuting variables, we present also a general Grobner bases theory for graded two-sided ideals of the graded algebra S = sum_i S_i with S_i = P σ^i and σ : P → P an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of P. Moreover, using a suitable grading for the algebra P compatible with the action of Σ , we obtain a bijective correspondence, preserving Grobner bases, between graded Σ-invariant ideals of P and a class of graded two-sided ideals of S. By means of the embedding ι this results in the unification, in the graded case, of the Grobner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials P = K [X × N] fits the proposed theory one obtains that, with respect to a suitable grading, the Grobner bases of finitely generated graded ordinary difference ideals can be computed also in the operators ring S and in a finite number of steps up to some fixed degree.