Details:
Title  Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra  Author(s)  Roberto La Scala, Viktor Levandovskyy  Type  Article in Journal  Abstract  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 Gröbner bases theory for graded twosided ideals of the graded algebra S = ⊕ iSi with Si = Pσi 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 Gröbner bases, between graded Σinvariant ideals of P and a class of graded twosided 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 noncommutative 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 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.  Keywords  Skew polynomial rings, Free algebras, Gröbner bases  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717112000934 
Language  English  Journal  Journal of Symbolic Computation  Volume  48  Number  0  Pages  110  131  Year  2013  Edition  0  Translation 
No  Refereed 
No 
