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 Title Skew Polynomial Rings, Grobner Bases and the Letterplace Embedding of the Free Associative Algera Author(s) Viktor Levandovskyy, R. La Scala Type Manual Abstract In this paper we introduce an algebra embedding $\iota:K< X >\to S$ from the free associative algebra $K< X >$ generated by a finite or countable set $X$ into the skew monoid ring $S = P * \Sigma$ defined by the commutative polynomial ring $P = K[X\times\N^*]$ and by the monoid $\Sigma = < \sigma >$ generated by a suitable endomorphism $\sigma:P\to P$. If $P = K[X]$ is any ring of polynomials in a countable set of commuting variables, we present also a general Gr\"obner bases theory for graded two-sided ideals of the graded algebra $S = \bigoplus_i S_i$ with $S_i = P \sigma^i$ and $\sigma:P \to P$ an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of $P$. Moreover, using a suitable $\N$-grading for the algebra $P$ compatible with the action of $\Sigma$, we obtain a bijective correspondence, preserving Gr\"obner bases, between graded $\Sigma$-invariant ideals of $P$ and a class of graded two-sided ideals of $S$. By means of the embedding $\iota$ this results in the unification, in the graded case, of the Gr\"obner bases theories for commutative and non-commutative polynomial rings. Finally, since the shift operators $x_i\mapsto x_{i+1}$ fits the proposed theory for $X = \{x_1,x_2,...\}$, one obtains also that Gr\"obner bases of finitely generated graded ordinary difference ideals can be computed by these methods in the operators ring $S$ and in a finite number of steps. Keywords Skew polynomial rings; Free algebras; Grobner bases Length 23 URL http://arxiv.org/abs/1009.4152 Language English Year 2010 Month October Translation No Refereed No
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