2010/09/21 by Roberto La Scala, La Scala, Roberto, Viktor Levandovskyy +1
Computer Science · Mathematics · #13P10 #16Z05 #68W30 #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1009.4152
openalex publication_date 2010/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce an algebra embedding \ι:K< X >\→ S from the\nfree associative algebra K< X > generated by a finite or countable set X\ninto the skew monoid ring S = P * \Σ defined by the commutative\npolynomial ring P = K[X\× N^*] and by the monoid \Σ = < \σ >\ngenerated by a suitable endomorphism \σ:P\→ P. If P = K[X] is any ring\nof polynomials in a countable set of commuting variables, we present also a\ngeneral Gr "obner bases theory for graded two-sided ideals of the graded\nalgebra S = bigoplusi Si with Si = P \σi and \σ:P \→ P an\nabstract endomorphism satisfying compatibility conditions with ordering and\ndivisibility of the monomials of P. Moreover, using a suitable grading for\nthe algebra P compatible with the action of \Σ, we obtain a bijective\ncorrespondence, preserving Gr "obner bases, between graded \Σ-invariant\nideals of P and a class of graded two-sided ideals of S. By means of the\nembedding \ι this results in the unification, in the graded case, of the\nGr "obner bases theories for commutative and non-commutative polynomial rings.\nFinally, since the ring of ordinary difference polynomials P = K[X\× N]\nfits the proposed theory one obtains that, with respect to a suitable grading,\nthe Gr "obner bases of finitely generated graded ordinary difference ideals can\nbe computed also in the operators ring S and in a finite number of steps up\nto some fixed degree.\n