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

In this paper we introduce an algebra embedding @i:K->S from the free associative algebra K generated by a finite or countable set X into the skew monoid ring [email protected][email protected] defined by the commutative polynomial ring P=K[XxN^@?] and by the monoid @S= generated by a suitable endomorphism @s: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 [email protected]?"iS"i with S"[email protected]^i and @s: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 @S, we obtain a bijective correspondence, preserving Grobner bases, between graded @S-invariant ideals of P and a class of graded two-sided ideals of S. By means of the embedding @i 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[XxN] 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.