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.
Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra
AbstractIn this paper we introduce an algebra embedding ι : K
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