Home | Quick Search | Advanced Search | Bibliography submission | Bibliography submission using bibtex | Bibliography submission using bibtex file | Links | Help | Internal


TitleSkew Polynomial Rings, Grobner Bases and the Letterplace Embedding of the Free Associative Algera
Author(s) Viktor Levandovskyy, R. La Scala
AbstractIn 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.
KeywordsSkew polynomial rings; Free algebras; Grobner bases
URL http://arxiv.org/abs/1009.4152
Translation No
Refereed No