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

摘要

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 * ∑ defined by the commutative polynomial ring P = K[X × N~*] and by the monoid ∑ = (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 = ⊕_i S_i with S_i = Pσ~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 E, we obtain a bijective correspondence, preserving Grobner bases, between graded 27-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 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.