Two-sided Gröbner Bases in Iterated Ore Extensions

Abstract

It is shown that finite Gröbner bases exist and can be computed for two-sided ideals of iterated Ore extensions (which are also called iterated skew polynomial rings) with commuting variables.Given a ring R consider an iterated Ore extension of R where the new variables commute with each other.Identifying the iterated Ore extension of R and the polynomial ring over R (in the same number of variables) as free left R-Modules all two-sided ideals of the iterated Ore extension are left ideals of the polynomial ring.We therefore define a Gröbner basis of a two-sided ideal of the iterated Ore extension as a Gröbner basis of this two-sided ideal regarded as a left ideal of the corresponding polynomial ring. This, of course, requires that left Gröbner bases exist in the polynomial ring.If there is an algorithm for computing a left Gröbner basis for any given finite subset of the polynomial ring this algorithm can be extended to compute two-sided Gröbner bases in the iterated Ore extension.Examples of ground rings R meeting this requirement are polynomial rings and solvable polynomial rings over fields or over principal ideal domains.Applications include solving the two-sided ideal membership problem and computing in residue class rings of two sided ideals.KeywordsPolynomial RingLeft IdealWeyl AlgebraGround RingResidue Class RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.