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

In this paper we introduce an algebra embedding $ι:K< X >\to S$ 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 $P = K[X\times N^*]$ and by the monoid $Σ= < σ>$ generated by a suitable endomorphism $σ:P\to P$. If $P = K[X]$ is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra $S = \bigoplus_i S_i$ with $S_i = P σ^i$ and $σ:P \to P$ 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 Gröbner 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 Gröbner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials $P = K[X\times N]$ fits the proposed theory one obtains that, with respect to a suitable grading, the Gröbner 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.